On the back of comfortably surviving in their first season in the Premier League, the Bees are looking to further strengthen their squad this summer.
Scottish full-back Aaron Hickey has already joined from Bologna, while Keane Lewis-Potter is on the brink of signing from Hull City on a long-term contract.
However, according to The Athletic, Brentford are closing in on beating the likes of Manchester United and Chelsea to Strakosha.
With his seven-year stint at Lazio having come to an end, the Albania international has been a notable name in the free agent category.
The 27-year-old has recently acknowledged that he would be interested in signing for a Premier League this summer, aware that several high-profile teams were in the market for a new stopper.
Fulham had been tipped to sign Strakosha at one stage before it was claimed that United and Chelsea were both preparing their own bids.
Nevertheless, it now appears that Brentford have made significant headway in the race to secure Strakosha's signature ahead of several bigger clubs.
As well as wanting to have a realistic chance of becoming first choice, the report adds that Strakosha is also keen to live in London.
Should he link up with Brentford, the long-term future of David Raya will naturally be questioned given that he is the established number one under Thomas Frank.
The 26-year-old has a contract until the end of 2023-24, leaving the Bees in no rush to part ways with one of their most prized assets.
However, having made 122 appearances in three years, the Spaniard will not expect to be demoted to the substitutes' bench after helping Brentford to 13th position during the last campaign.
Raya also made his debut for his country as recently as March, a further sign that his reputation has never been higher than it is after his first year in the Premier League.