The married Labour MP allegedly paid for the services of two male escorts, discussed using the drug “Poppers” and offered to pay for a Class A drug, according to the Sunday Mirror.
The committee is currently considering a review of prostitution laws. Mr Vaz also opposed government attempts to criminalise Poppers.
In a statement, Mr Vaz said he will inform other members of the Home Affairs Select Committee whether he will stand aside when it meets on Tuesday.
Mr Vaz said: “It is deeply troubling that a national newspaper should have paid individuals who have acted in this way.
“I have referred these allegations to my solicitor Mark Stephens of Howard Kennedy who will consider them carefully and advise me accordingly.
“At this time I do not want there to be any distraction from the important work the Home Affairs Select Committee undertakes so well.
“Select committees do vital work in holding the government and others to account. We are due to publish two reports, one into anti-Semitism and the other into FGM in the next few days, in addition we have a number of key witnesses.
“I will of course inform committee members first of my plans when we meet on Tuesday.
“My decision has been based entirely on what is in the best interests of the committee which I have had the privilege of chairing for the last nine years.”
Select committee member Naz Shah, who has spoken to Mr Vaz, has told Sky News “he has done the right thing” by standing aside as chair, but Mr Vaz has yet to confirm that is the case.
Mr Vaz has been the Labour MP for Leicester East since 1987 and chair of the Home Affairs Select Committee for 10 years.
In July, the Home Affairs Select Committee issued a report saying that soliciting by sex workers should be decriminalised.
A Labour Party spokesperson said: “Keith Vaz has issued a statement on this matter.
“As with all departmental select committees, Keith was elected to the chair of the Home Affairs Select Committee by the House of Commons, and his position is a matter for him and the House.”
Former culture secretary John Whittingdale said that Mr Vaz’s reported resignation seemed “sensible”.
“I haven’t read the whole of the allegations and therefore it’s difficult to comment on them,” he told Sky News’ Murnaghan programme.
“But Keith Vaz as I understand it is that he will stand aside from the chairmanship of the select committee.
“Given the areas of which the committee is responsible, that does seem to me to be a sensible course of action. I wouldn’t want to comment beyond that.”