Hancock follows former foreign minister Boris Johnson, who is seen as the favorite, current foreign minister Jeremy Hunt, International Development Secretary Rory Stewart and former work and pensions minister Esther McVey in joining the contest.
"I'm going to run to be the next prime minister because I believe from the bottom of my heart we need a leader for the future not just for now," Hancock told BBC radio.
"Of course we have to deliver Brexit and I will."
May said on Friday she would step down as Conservative Party leader on June 7 after admitting defeat in her bid to get parliament to pass a divorce deal she agreed with the European Union (EU), Reuters reported.
The opposition Labor Party said it wanted an election as soon as possible. Labor's finance spokesman John McDonnell confirmed on Saturday the party would bring a vote of no confidence in the government if it looked like it might pass.