He has given up his chairmanship of the Home Affairs Committee following the allegations.
The Mirror newspaper published new transcripts of a recording of the MP talking with the two escorts, which it said scotched suggestions that he was the victim of a “press sting”.
According to the paper, the tape showed Mr Vaz telling one of the men to treat the other as his “bitch” and discussing having sex.
Following reports in the Sunday Mirror at the weekend that he had paid escorts and discussed the use of legal and illegal drugs, the Leicester East MP initially told reporters he planned to stand aside as chairman “with immediate effect”.
But he later said publicly that he wanted to discuss the situation with committee members before deciding how to proceed.
Prime Minister Theresa May stepped up pressure when she said that voters must be able to have confidence in their politicians.
Speaking during a visit to China, Mrs May said: “What Keith does is for Keith and any decisions he wishes to make are for him.”
But she added: “I have always been clear throughout my political career that what is important for people is that they feel that they are able to have confidence in their politicians. That is what I think we all have a duty to provide for those who elect us.”
Asked whether he expected Mr Vaz to step down, former Labour minister Ed Balls told ITV1’s Good Morning Britain: “I think he will. I think in the end, yes.”
Mr Vaz, who has publicly apologised to his wife and children for causing them “hurt and distress”, is facing a possible investigation by Commons’ sleaze watchdogs.
The Sunday Mirror claimed that money was paid into an account used by one of the escorts by a man linked to a charity set up by the MP, though there was no suggestion that the charity’s money was used.
Labour leader Jeremy Corbyn indicated there was no need for Mr Vaz to stand down from the 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.”