UPDATE: From what I am picking up, this is more than a rumor at least in the effect if not in the exact verbiage.

From the Italian site La Nuova Bussola Quotidiana, which has an English page as well as Italian. This is from the English side for convenience. My emphases and comments.
The Pope: “Away with Cardinal Burke’s house and salary”.
Vatican sources close to the Daily Co
mpass: Burke was defined as an “enemy” in an announcement made to the Heads of Dicasteries of the Roman Curia. The cardinal has not yet received a formal notice, but considering precedents, it is unlikely to be just a threat, which nonetheless would be very serious.
“Cardinal Burke is my enemy, so I am taking away his flat and salary”. This is what Pope Francis supposedly said at the meeting with the Heads of Dicasteries of the Roman Curia last 20 November, and which a Vatican source revealed to the Daily Compass. The indiscretion [in Italian another word for a “revelation”] was later confirmed by other sources. As far as we are aware, Cardinal Raymond L. Burke, currently in the United States, has not yet received a formal notice confirming the Pope’s words, but given the precedents – most recently the case of Monsignor Georg Ganswein, former personal secretary of Pope Benedict XVI – there is little doubt that words will be followed by deeds. Nor would the difficulty in canonically justifying such a measure be an obstacle, given the contempt for the laws of the Church also shown by Pope Francis on the occasion of the removal of bishops from their dioceses (see here).
The alleged enmity of Cardinal Burke has become a real obsession for Pope Francis in recent times, but in reality the American cardinal has been in the crosshairs since the beginning of his pontificate, probably because he encapsulates some of the elements that most annoy him: he is American and is a constant reminder of the doctrine and Tradition of the Church; and in addition he resides in Rome, a stone’s throw from St Peter’s Square, from where – the Pope will think – he can ‘plot’ against him. [Unless you are certain type of American….]
Certainly Burke has been very clear in his criticism of the concept of synodality, which has now become a mantra intended to change the nature of the Church, and at the conference ‘The Synodal Babel’ on 3 October last, organised in Rome by La Bussola [I was there.] precisely on the eve of the opening of the Synod on Synodality, his arguments and his direct polemic with the new Prefect of the Dicastery for the Doctrine of the Faith, Victor “Tucho” Fernandez, who had called Cardinal Burke a heretic and schismatic and those who ask the Pope to “safeguard and promote the depositum fidei”, had made a lot of noise. After all, calling the Pope to task is part of the cardinals’ duty, and Francis himself has repeatedly encouraged (in words) parousia. [sic. The word is parrhesia – “frankness, boldness of speech”.]
And Cardinal Burke has always strongly rejected the label of “enemy of the Pope” that they have wanted to stick on him since the beginning of the pontificate, especially since he criticised the position of Cardinal Walter Kasper who, in preparation for the 2014 Synod on the Family, explicitly called for access to communion for remarried divorcees. Burke was in good company, yet especially against him a real campaign of demonisation was focused, painted as the director of occult plots against Pope Francis (accusations that Burke has always strongly rejected). [“director of occult plots”… “registra di trame occulte”… a better translation would be “orchestrator of secret/hidden plots”. This doesn’t involve, “the occult”.]
[… lists ways in which Francis as gone after Burke over the years… ]
In the meantime, however, in recent years Pope Francis has never missed an opportunity to launch personal jibes at Cardinal Burke, reaching a climax with the unfortunate (to put it mildly) joke uttered while Cardinal Burke was struggling between life and death because of Covid: “Even in the College of Cardinals there are some deniers,” the Pope said with a satisfied smile in the press conference on the plane returning from his trip to Hungary and Slovakia on 15 September 2021, “and one of them, poor man, is hospitalised with the virus.”
The second round of Dubia, presented last July together with Cardinals Brandmüller, Sarah, Zen and Sandoval, but only made public on 2 October, will undoubtedly have irritated the Pope even more, who seems to have let go of his inhibitions after the death of Benedict XVI last January. Thus the new prefect of the Dicastery for the Doctrine of the Faith, Fernandez, was able to personally target Cardinal Burke in the aforementioned interview with the National Catholic Register in September in what, in retrospect, can be considered a warning.
And now here we come to the Pope’s announced decision to strike Cardinal Burke directly, taking away his flat and salary, a serious and unprecedented measure, in defiance of every legal and ecclesial principle. One may think that the real purpose is to remove Burke from Rome, weakening the camp of those who resist the revolution in progress, as a Conclave approaches, but it is also a warning to those who work in the Roman Curia. The fact is that the end of this pontificate increasingly resembles in its methods, a South American dictatorship.
On the other hand… wasn’t there a story some time ago that Francis was planning on taking away the residences of all the non-Curial cardinals in Rome? It could be that this present story is the source of that rumor.
On the other other hand… I suspect that this is sort of a “Brer Rabbit” moment. “Please, Brer Francis! Don’t throw me into that briar patch!”
The Briar Patch in this case being the place where Card. Burke probably longs the most to be….

NOTA BENE:
I will be rather strict with the comment moderation queue. So, think about it first. Going full oooga-booga isn’t going to get approved.