International Trade Minister Penny Mordaunt was the latest to announce her bid, saying the UK "needs to become a little less about the leader and a lot more about the ship".
Former health secretaries Sajid Javid and Jeremy Hunt have also joined one of the most open leadership races in recent history.
Other contenders include frontrunner Rishi Sunak, the former Treasury chief, and Nadim Zahawi, who took Sunak's job as chancellor of the exchequer last week. Foreign Secretary Liz Truss hasn't formally announced but is expected to join the race.
The race comes after Johnson, 58, was brought down by a series of scandals, the most recent involving his decision to promote a lawmaker who had been accused of sexual misconduct to a senior position in his government.
An influential Conservative party committee is expected to lay out the rules for the leadership contest this week, with news reports suggesting that Conservative lawmakers will narrow the field to two before Parliament breaks for its summer recess on July 21.
Party members around the country will then vote on the final choice before the end of August, the Times of London reported.
Johnson has said he will remain prime minister until his successor is chosen. But many want him to go now, with even some Conservative politicians worried that he could do mischief even as a caretaker prime minister.
As politicians took to the airwaves to endorse candidates, many sought to distance their favourites from the turmoil of the Johnson years by stressing traits such as "integrity" and "honesty".
Karan Bilimoria, the former president of the Confederation of British Industry, said the decision should be made as quickly as possible because businesses that are still struggling to overcome the impact of the pandemic and are now facing the growing possibility of a recession need help now.
"We have got to get through this period as quickly as possible and find a good leader who can then rebuild trust," he told Times Radio. "It is rebuilding the trust with the country as well. The country has lost that trust and business is very worried."