The 30-year-old made the move to Ewood Park from Belgian outfit Gent in August 2020 and has since made 118 appearances for the club, although his first-choice status has been thrown into doubt in recent times.
A knee injury saw the Belgium international miss two months during the second half of the 2022-23 Championship term, and Kaminski's deputy Aynsley Pears did enough to keep his place in between the sticks for the rest of the campaign.
The 2-2 draw with Birmingham City in late January was the last time the former Copenhagen stopper played for Rovers, and that match at Blackburn's Lancashire home could prove to be his last.
According to the Lancashire Telegraph, Luton Town have seen multiple bids for Kaminski turned down by Blackburn during this summer's transfer window, as the top-flight newcomers seek to bolster their goalkeeping options.
The Hatters gained promotion to the Premier League via the playoffs last term, beating Coventry City in a dramatic penalty shootout at Wembley in late May.
Key to Luton's rise to the top flight was on-loan keeper Ethan Horvath, however the American international will not be returning to Kenilworth Road and is now set to link up with his parent club Nottingham Forest.
That decision leaves Luton in need of a replacement between the sticks, and it is understood that of all the Hatters' bids up to this point - all failing to reach the seven-figure mark - have been rejected by Blackburn.
It is believed that Rovers are hoping to pocket between £3m-£4m for the sale of Kaminski, with the 30-year-old still having two years left on his contract in Lancashire.
After joining from Gent for around £450,000 in 2020 from Gent, Jon Dahl Tomasson's side could be set to make a significant profit on the keeper as they look to build a squad capable of reaching the top six next season.
The report adds that a deal for Kaminski is not close to completion due to the two club's disagreement over a suitable transfer fee, however it is said that Luton are not ready to end their efforts to lure the American to Kenilworth Road.
Rob Edwards has already recruited from the second tier of England during this window, bringing in Ireland international Chiedozie Ogbene from Rotherham United to bolster the squad's attacking firepower.
Mads Andersen has also joined from League One club Barnsley for an undisclosed fee, with the Dane likely to form a partnership with Tom Lockyer at the heart of the Hatters' defence after the 28-year-old signed a new deal at the club.
Despite transfer speculation, Kaminski has travelled to Austria with his Rovers teammates as they prepare for their fifth season in the Championship since earning promotion from the third tier in 2018.