Weghorst joined Burnley from Wolfsburg for £12m in January 2022 and mustered two goals in 20 Premier League appearances as the Clarets suffered Premier League relegation.
The Dutchman went on to spend time on loan at Besiktas, Manchester United and Hoffenheim, before he returned to Turf Moor for the start of the current season.
Weghorst featured as a substitute for Burnley's first two matches, but he was not involved in games against Sunderland and Wolverhampton Wanderers in the league and EFL Cup respectively.
The 32-year-old has now left the Clarets to complete a permanent move to Ajax where he has signed a two-year contract until June 2026.
Ajax technical director Alex Kroes told the club's official website: "Wout stands for an absolute top sports experience, and we find that extremely important at Ajax.
"He is a prolific striker who constantly makes things difficult for every opponent. There are many matches coming up in the next few months, so we are happy that Wout is joining us."
Having been unveiled as an Ajax player, Weghorst will have to wait until after the international break to make his debut after Ajax's meeting with Feyenoord was cancelled due to a planned police strike.
Celtic agree Engels transfer
Meanwhile, in the Scottish Premiership, Celtic have reportedly reached an agreement to sign Augsburg midfielder Arne Engels in a club-record deal.
Celtic have seemingly identified Engels as a replacement for Matt O'Riley, who recently completed a move to Brighton & Hove Albion.
After failing with their first three bids, Celtic have made a breakthrough and are now on the verge of unveiling the 20-year-old as their latest summer addition.
According to Sky Sports News, Celtic have reached an agreement to sign Engels in a club-record £11m deal, surpassing the £9m they paid Paris Saint-Germain to recruit Odsonne Edouard.
Engels has agreed a four-year contract with the Hoops and is set to travel to Glasgow on Friday to undergo a medical.
The Belgian midfielder will leave Augsburg after contributing three goals and nine assists in 53 appearances in all competitions.
Watford set to sign Jebbison
In the Championship, Watford are on the verge of completing the loan signing of Bournemouth striker Daniel Jebbison.
The Cherries signed Jebbison on a free transfer after the striker left Sheffield United upon the expiry of his contract at the end of last season.
Jebbison has featured in all three of Bournemouth's competitive matches this season, appearing as a substitute against Nottingham Forest and Newcastle United before playing the first half of Wednesday's EFL Cup defeat to West Ham United.
However, their plan has always been for Bournemouth to send the 21-year-old out on loan to gain regular playing time.
According to The Athletic, Jebbison is on the verge of joining Watford on a season-long loan deal after undergoing a medical.
Watford boss Tom Cleverly has confirmed that Jebbison has already linked up with his new teammates, leaving him in line to play some role in Saturday's Championship clash with Sheffield United.