Vaz stepped down as chair of the Home Affairs Select Committee yesterday after the Sunday Mirror reported the 59-year-old MP for Leicester East paid for the men to visit him at his flat in London just a week ago. It also claimed the MP offered to pay for drugs for the men.
Vaz, who chaired the committee from 2007, was leading the committee through inquiries in areas such as prostitution.
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As part of parliamentary rules, the committee must be chaired by a Labour MP, with chairs elected by a vote of MPs.
And now Umunna has decided to seek the leadership of the watchdog.
City A.M. understands the Labour MP is keen to ensure continuity of work currently being undertaken by the group, which will include reports into anti-semitism and female genital mutilation, both of which are due to be released shortly.
In addition, it is understood the Streatham MP had noted that Vaz was the only black or minority ethnic chair of any of parliament's select committees prior to his resignation.
Former shadow home secretary Yvette Cooper has also been linked with a bid to chair the committee.
It would mark a return to the limelight for Cooper, who has sat on the backbenches since serving in Ed Miliband's shadow cabinet.
Tory MP Tim Loughton will serve as acting chair of the committee until a Labour MP is elected after parliament returns from recess for party conferences.