The Sunday Mirror alleged Mr Vaz met the men at his flat in north-west London on 27 August.
In a statement, Mr Vaz said: “It is deeply disturbing that a national newspaper should have paid individuals to 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.”
Meanwhile, Labour leader Jeremy Corbyn described the allegations as a “private matter”. Mr Corbyn said: “He [Keith Vaz] is going to meet the home affairs select committee and discuss with them what his role will be in the future, I’m not sure what his decision will be, I will leave it to him to decide on that.
“He has not committed any crime that I know of, as far I’m aware it is a private matter.”
However, in a separate statement issued to the Mail on Sunday, the Leicester East MP said: “I am genuinely sorry for the hurt and distress that has been caused by my actions, in particular to my wife and children.
“I will be informing the committee of my intention to stand aside from chairing the sessions of the committee with immediate effect.”
He later issued another statement, which added: “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 antisemitism and the other into FGM in the next few days, in addition we have a number of key witnesses.
“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 nineyears.”
Naz Shah, a member of the committee, said she had spoken to Mr Vaz, adding “he has done the right thing” by standing aside.
A Labour Party spokeswoman 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.”
The home affairs committee is currently examining a review of prostitution laws.
Former culture secretary John Whittingdale said that Keith Vaz’s reported resignation seemed “sensible”.
“I haven’t read the whole of the allegations and therefore it’s difficult to comment on them,” hesaid.
“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.”
Labour shadow health secretary Diane Abbott said this was a “dreadful” experience for Mr Vaz, and his wife and children.
“I have known Keith for over 30 years, I think this must be a dreadful time for him, and his family – his wife and his two children. And I would rather not comment,” she said.
Mr Vaz, who is married with two children, was first elected as an MP in June 1987. He was elected chairman of the home affairs select committee in 2007.
The committee is next due to sit at 3:30pm tomorrowwhen it is expected to hear evidence on the work of the immigration directorates.