The striker has interest from the Premier League but is likely to remain at Carrow Road beyond the end of the window next week, the PA news agency understands.
Wolves, Leeds, Aston Villa, West Brom and Sheffield United have all enquired about the Finland international this summer.
Pukki also has interest from Germany and Turkey but there have been no official approaches with the expectation Norwich’s asking price will be too high.
He also turned 30 this year meaning he will have little resale value which has also been an issue for some clubs.
But the Canaries will need to be challenging for promotion in January or they could lose him with the forward harbouring top-flight ambitions.
He signed a three-year deal 12 months ago and, while he is in no rush to leave Norfolk and is happy at Carrow Road, Pukki proved himself in the Premier League last term.
He scored 11 goals as Daniel Farke’s side finished bottom of the table.
Pukki has netted 42 times in 86 appearances for the Canaries since joining on a free transfer in 2018 and scored 29 goals when they won the Sky Bet Championship in 2018-2019.
Norwich’s confidence of keeping their star players, including Ben Godfrey and Todd Cantwell, has grown and they have not had any bids.
They are prepared to sell Emi Buendia, with the attacking midfielder wanting to stay in the Premier League, and could still struggle to keep Max Aarons after Barcelona’s interest.
Jamal Lewis joined Newcastle for an initial £15million this month but the Canaries are under no financial pressure to sell and tied many of their stars to new deals last summer.
Josip Drmic, Tom Trybull and Moritz Leitner are not needed with midfielder Leitner talking to three Championship clubs.
