The former England cricketer, who now lives near Richmond in North Yorkshire, 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.
There will be peerages also for Ian Austin and Sheffield-born 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All three had the Tory whip withdrawn after seeking to block his efforts to reach an agreement with Brussels.
Sir Ian – one of England’s greatest cricketers with 5,200 test runs and 383 wickets – publicly supported the Brexit campaign, appearing alongside Mr Johnson before the referendum.
There was no immediate comment from Downing Street.
There was also no answer at the cricket star’s £2 million gated home, near Richmond where it was understood he had held a cook-along charity barbecue the night before.