The Sun says the Premier League new boys are now poised to firm up their interest with an official bid.
Ajer only has 12 months left on his Celtic deal — but could still command a fee of around £10million.
Newcastle United are also keen but Norwich are now stepping up their interest in the 23-year-old.
Director of football Stuart Webber saw Ajer in the final weeks of the season as he tweaks the squad for the Premier League.
Norwich will reinvest some of that cash, and a new centre-half is one of boss Daniel Farke's top priorities.