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The former universities minister is the 13th candidate to join the race, which will also choose the UK's next PM.
Mr Gyimah - who quit over Theresa May's Brexit plan - said he would vote Remain in such a poll, but would not "actively campaign" if he became prime minister.
Meanwhile, other contenders to succeed Mrs May have been setting out their Brexit plans.
Former Commons Leader Andrea Leadsom said she would seek a "managed exit" by 31 October - the deadline the EU has set for leaving the bloc.
She told the BBC's Andrew Marr Show that Mrs May's negotiated plan was "dead", as the EU would not re-open it and Parliament would not vote for it.
Mrs Leadsom said she would instead introduce legislation to guarantee citizens' rights, ramp up preparations for all Brexit scenarios and explore alternatives to the Irish border backstop plan.
Home Secretary Sajid Javid, who is also running in the Tory leadership race, said he "passionately" wanted to leave the EU with a deal, but it was "responsible" to prepare for a no-deal exit.
His pitch for the top job was a "modern digitised border" between Northern Ireland and the Republic of Ireland, which he said would "unlock a Brexit deal".
Appearing on the Andrew Marr Show, Mr Javid said he would try to renegotiate the Irish border backstop plan and offer to pay Ireland for work towards a digital border, saying it was "morally right".
He also said there should not be another general election before Brexit is delivered, but admitted he "may not be able to stop it" if the government were defeated in a vote of no confidence. [...]