The former England cricketer is among 30 new peers to be announced later this month, according to The Times.
The list is said also to include four ex-Labour MPs – including Frank Field and Gisela Stuart – who both supported Brexit in the 2016 referendum.
The Times report all the former Labour MPs are expected to sit as independents in the Lords, adding that none of Jeremy Corbyn’s three nominations are included.
There will be peerages also for Ian Austin and John Woodcock, who backed Remain but who supported Mr Johnson’s Brexit deal and urged Labour voters not to support Jeremy Corbyn, the paper said.
At the same time, it was reported that Mr Johnson will seek to heal the divisions in the Conservative Party with peerages for Ken Clarke, Philip Hammond and Ed Vaizey.
All three had the Tory whip withdrawn after seeking to block his efforts to reach an agreement with Brussels.
The list will confirm that none of Mr Corbyn’s three controversial nominations — John Bercow, Karie Murphy and Tom Watson —
Sir Ian – one of England’s greatest cricketers with 5,200 test runs and 383 wickets – publicly supported the Brexit campaign, appearing alongside the PM.
The 64-year-old gave his backing to Boris Johnson ahead of the EU referendum, writing in The Sunday Times: “Cricket is a game where you achieve the greatest success when you are confident in your own ability to go out and stand proud. Britain has that spirit.”
Adding: “It is insane that because we are in the EU we do not have the freedom to reach our own agreements to trade freely with these places or with emerging powers like China.
“We have lost the right to govern ourselves, to make our own laws and to choose who comes here.”
Last year Boris Johnson heaped praise on Sir Ian when he compared himself to the ‘swashbuckling’ Botham to Theresa May’s steadfast cricketing hero Geoffrey Boycott.
Speaking at a hustings event in June last year, Mr Johnson was asked if he would be constrained as a PM, replying: “I think that there is one way to do this thing now, if I may venture a cricketing metaphor, I think we’ve had quite a lot of Boycott on the wicket and it is time for Botham to come in. That is my view,
“Particularly in the EU negotiations, we cannot have the same old, same old. We cannot have a can-kicking approach. We kick the can, we will kick the bucket, we’ve got to get on and do this.”
There was no immediate comment from Downing Street.