The high-profile Labour MP indicated he will stand aside as chairman of the powerful Commons Home Affairs Committee at a meeting on Tuesday after publicly apologising to his wife and children for the “hurt and distress” he caused them.
Mr Vaz met the escorts at a flat he owns near his family home in North London last month, according to reports in the Sunday Mirror.
Tory MP Andrew Bridgen told the Press Association he would write to the Parliamentary Commissioner for Standards regarding Mr Vaz’s conduct.
The Sunday Mirror also claimed money was paid into an account used by one of the escorts by a man linked to a charity set up by the MP.
The Charity Commission said the regulator would consider becoming involved in the matter if it felt such action was needed.
A spokesman told the Press Association: “The Charity Commission is aware of the allegations made regarding an individual linked to the charity Silver Star.
“The commission has asked journalists to forward any details and evidence they have in order to determine whether there is a regulatory role for the commission.”
There is no suggestion in the newspaper report that the charity’s money was used in the affair.
The Sunday Mirror also claimed that Mr Vaz had a conversation regarding cocaine with one of the escorts in which the MP said he did not want to use the drug, but indicated he would pay for it for the other man at a later date.
Mr Vaz also asked the escort to bring party drug poppers with him for the encounter, according to the report.
Labour leader Jeremy Corbyn indicated there was no need for Mr Vaz to stand down from the Labour Party, stating: “Well, he hasn’t committed any crime that I know of. As far as I’m aware it is a private matter, and I will obviously be talking to Keith.”
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.
“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 (female genital mutilation) 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.”
In an earlier 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 on Tuesday of my intention to stand aside from chairing the sessions of the committee with immediate effect.”
Naz Shah, a member of the Home Affairs Committee, told Sky News she had spoken to Mr Vaz, adding “he has done the right thing” by standing aside.