Manager David Moyes has personally kept tabs on the Newcastle United academy graduate as he looks to replace Sebastien Haller ahead of next season.
The Ivory Coast international striker was sold to Ajax last month, but the Hammers could not get a replacement before the transfer window shut and will end the campaign with Michail Antonio as their only natural number 9.
Armstrong has bagged 20 goals and three assists in 34 games across all competitions this term, but Rovers are 12 points off the promotion play-offs spots and could be forced to cash in on him in the coming months.
The 24-year-old is set to be a free agent in the summer of 2022, and West Ham were hoping that £10 million would be enough to land him.
However, with Newcastle due 40 percent of whatever amount the Championship outfit sell him for as a result of a sell-on clause, Blackburn have decided to slap a huge asking price on him, believing they can get something close to the £28 million Brentford earned when they sold Ollie Watkins to Aston Villa last summer.
The Hammers have to get a quality striker this summer, and Armstrong could prove to be an excellent addition.
Whether West Ham would be keen to meet the demands of his club remains to be seen, though.
A move for Brentford’s Ivan Toney could also be on the cards should the second-placed Championship outfit fail to secure promotion again.
Toney, now rated £30 million, joined Brentford on a five-year deal last summer after ending his two seasons with Peterborough United with 49 goals.
He has hit the ground running in the Championship, scoring 24 goals and assisting nine others this season.