The 22-year-old defensive midfielder moved to the San Siro from the fellow Serie A side Atalanta two years ago.
The versatile midfielder made 34 matches in Serie A last campaign and netted seven goals as AC Milan finished fifth but missed out on UEFA Champions League football.
It was a two-year loan deal with Rossoneri obliged to buy the Ivory Coast international for around £24 million this summer.
Now it seems the Italian giants may land Kessie but immediately sell the player on to help balance the books.
Arsenal are closely monitoring the current situation, but any possible move for Frank could see them use a big chunk of their summer kitty to bag the midfielder as the San Siro outfit themselves have to watch the pennies.
Failure to qualify for the UEFA Champions League has proven costly for Spanish manager Unai Emery.
He has just £40m to work with to improve his roster in order to challenge for silverware next campaign.
Unai may need to sell before such a big outlay, allowing Tottenham Hotspur or West Ham United to steal a march on Arsenal.
Frank could be a replacement for English talent Declan Rice who is high on Ole Gunnar Solskjaer’s Manchester United’s shopping list.
And if he departs Italy, Frank wants to play European football, a demand that could scupper any possible Hammers move.