Vaz, 59, who is expected to announce his decision to step aside as chair of the influential Commons' Home Affairs Committee tomorrow, is to be referred to the UK's Parliamentary Commissioner for Standards Kathryn Hudson by Conservative MP Andrew Bridgen.
"We need a full police and criminal investigation into Mr Vaz's affairs, I think we need the Parliamentary Standards investigation...At the end of the day I think Mr Vaz has historically and is now currently bringing Parliament into disrepute and I don't think he is a fit and proper person to be a Member of Parliament," Bridgen said.
He is alleged to have told the escorts to bring the party drug known as "poppers" and is also quoted as discussing the possibility of paying for cocaine at a future meeting, but added that he would not take the drug himself.
"It should be treated as a private matter. He is going to meet the Home Affairs Committee and discuss with them what his role will be in the future. I'm not sure what that decision will be but I leave it to him to decide on that," said Labour party leader Jeremy Corbyn, who appeared to indicate the party had no intention to take any further action against Vaz.
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A Downing Street spokesperson said today that it was "up to him" if he feels his chairmanship of the Home Affairs Committee is still tenable.
While Vaz has indicated his decision to "step aside" from the role, a formal announcement on whether this is a temporary or permanent move is expected only tomorrow.
He is accused of meeting the escorts at least twice last month and is alleged to have told them that his name was Jim, adding that he was a washing-machine salesman.
Malde Modhwadia, who has served as a trustee of the charity, said he was unaware of the payments or whether the charity had any involvement.