The 31-year-old is currently with Netherlands at Euro 2024, and he scored his country's winner in their opening match of the summer competition against Poland.
The experienced striker spent the 2023-24 campaign on loan at Hoffenheim, scoring seven goals and registering four assists in 30 appearances in all competitions.
Weghorst is now due to return to Burnley, where he has a contract until June 2025, but the Dutchman is widely expected to leave the Clarets on a permanent basis this summer.
According to The Telegraph, Ajax are considering making a move for the striker this summer, with Burnley thought to be willing to accept a cut-price fee for his services.
Weghorst famously spent the second half of the 2022-23 season on loan at Manchester United, scoring twice and registering three assists in 31 appearances for the Red Devils.
Ipswich 'make fresh bid for forward Ioannidis'
Elsewhere, Ipswich Town have allegedly had a fresh €25m (£21.1m) offer for Fotis Ioannidis rejected by Panathinaikos.
The 24-year-old scored 23 goals and registered nine assists in 44 appearances for his Greek club during the recent campaign, and a number of clubs are thought to be keen on his services.
According to Sport24, Ipswich recently submitted a renewed €25m (£21.1m) bid for the forward, having seen their opening offer for his services rejected.
However, the report claims that the newly-promoted Premier League outfit will have to come up with more money if they wish to sign the Greece international.
Ioannidis has allegedly been identified by Ipswich as a priority target for the summer transfer window, but Lazio and Genoa are also battling for his signature.
Ipswich's initial offer for the forward is believed to have been around €22.5m (£19m).
Sheffield United 'open talks with Forest over Worrall'
Meanwhile, Sheffield United have allegedly opened discussions with Nottingham Forest over a deal for central defender Joe Worrall during this summer's transfer window.
The Blades are said to be long-term admirers of Worrall and are believed to have made an attempt to sign him in January, only for their move to be knocked back by the Reds.
According to TEAMtalk, the recently-relegated outfit are now in talks with Forest in the hope of coming to an agreement over a deal during the current market.
Worrall spent the second half of last season on loan at Turkish outfit Besiktas, representing the club on nine occasions in all competitions, scoring once.
The 27-year-old has turned out for Forest 226 times since breaking into the first team, meanwhile, and he has another two years left to run on his contract at the City Ground.