The 28-year-old has been a regular fixture during a largely-miserable campaign for the Clarets, scoring four times and assisting twice in 30 Premier League matches.
However, Brownhill has played second fiddle to Sander Berge and Josh Cullen in the engine room of late, being restricted to minutes as a substitute.
That being said, the Englishman is still managing to come up with vital contributions for Vincent Kompany's troops, netting the opener in Burnley's 1-1 draw with Brighton & Hove Albion on Saturday afternoon.
Since making the switch from Bristol City to Turf Moor in January 2020, the midfielder has featured 164 times for the Clarets, scoring on 14 occasions.
West Ham dealt blow in Brownhill pursuit?
According to HITC, Burnley are prepared to trigger the extension clause in Brownhill's deal at Turf Moor, changing the landscape of his potential exit in the summer.
The midfielder's current deal with the Clarets expires at the conclusion of the 2023-24 campaign, meaning that he would be available on a free transfer.
However, with Burnley supposedly set to extend Brownhill's contract by another year, any interested parties - such as West Ham - will need to pay a transfer fee for his services.
Having forked out in the region of £9m for the 28-year-old from Bristol City in 2020, it is understood that Kompany's side want to squeeze as much as possible out of the possible sale of the player this summer.
David Moyes's men were in the race for Brownhill during the January window of 2020, however the former Preston North End man opted for a switch to Turf Moor instead, citing the Hammers' precarious position in the Premier League at the time as a reason for his decision.
Crystal Palace, Brentford join West Ham in Brownhill race
On top of the Europa Conference League holders, it is understood that Crystal Palace and Brentford are also keen on Burnley's Brownhill ahead of the summer trading point.
Despite securing the services of highly-rated youngster Adam Wharton from Blackburn Rovers during the winter window, it is said that the Eagles are still targeting reinforcements for their midfield.
Oliver Glasner's men stunned the footballing world on Sunday afternoon, when they visited Anfield and secured a 1-0 win, derailing Liverpool's ambitions of lifting the Premier League title.
On the other hand, Brentford are continuing to struggle during their third-straight campaign in the top flight, however Thomas Frank's men recorded a much-needed 2-0 victory over Championship-bound Sheffield United over the weekend.