The 25-year-old was heavily linked with a move away from his Dutch club during last summer's transfer window, with Chelsea thought to have been the favourites for his signature.
Alvarez ultimately remained in Amsterdam for the 2022-23 campaign, and he was again a key player for his club, making 44 appearances in all competitions, contributing four goals and three assists in the process.
The Mexico international has another two years left to run on his contract at Ajax, but there has been a host of speculation surrounding his future in recent months, with Dortmund seemingly emerging as the favourites to purchase him ahead of the 2023-24 season.
However, according to De Telegraaf, a move has broken down over Ajax's asking price, which Dortmund are not prepared to meet.
The report claims that the Dutch giants want between €30m (£25.8m) and €40m (£34.4m) for the Mexican, but Dortmund view the deal as too expensive and have therefore dropped their pursuit of him.
Alvarez, who has also been linked with a reunion with Erik ten Hag at Manchester United, made the move to Ajax from America in the summer of 2019.
The midfielder has featured on 147 occasions for his current side in all competitions, contributing 13 goals and six assists in the process.
Alvarez has also been capped on 63 occasions by his country, and it is still thought that he will leave Ajax this summer despite Dortmund allegedly ending their interest.
Last month, the midfielder said that he was ready to make the next step in his career.
"I spent four wonderful years here and this place still feels like home to me. I won a lot of trophies and developed. This could be the right time to take the next step in my career. But everything has to be perfect. I need to improve, but so do Ajax," he told Voetbal International.
"My entourage lets me know when a club's interest is concrete, only then do I think about it myself. I also read and hear about the interest in the media.
"Of course it's nice to be linked with a big club, but Ajax is also a big club. So I will always show respect to everyone in and around Amsterdam. I am very grateful to everyone here."
Dortmund have sold Jude Bellingham to Real Madrid this summer in a big-money deal, while Ansgar Knauff, Raphael Guerreiro, Mahmoud Dahoud, Felix Passlack, Luca Unbehaun and Anthony Modeste have also moved on ahead of the 2023-24 campaign.