The newspaper said Johnson, as his main rival, Interior Minister Theresa May, will announce their final plans this week.
"Boris Johnson summoned several Tory MPs to his home today has he prepares to mount an expected leadership bid", — the newspaper writes.
UK Home Minister Theresa May will in the near future also nominate her candidacy for the chief post, local media reported late Sunday.
According to The Telegraph newspaper, May has already gathered a team which is aimed to help her in the party leadership race.
A committee of Conservative members of parliament is due to meet on Monday to agree on the rules of choosing a party leader, with a shortlist of just two candidates expected by the end of July. The new leader will be voted on in September and announced at the party conference in October.
Tories are preparing to vote on a new prime minister by October to replace David Cameron, who declared his resignation on Friday after the UK's vote to leave the European Union.
The polls have now closed, democracy has been served + we await the verdict of the people. Thanks to everyone involved + everyone who voted— Boris Johnson (@BorisJohnson) June 23, 2016