According to transfer journalist Florian Plettenberg, the Bundesliga side have increased their bid to €5m, though Ajax are yet to give Frankfurt an answer.
The German club's opening move had gone down badly in Amsterdam. Frankfurt initially tabled an offer of just over €2m for the Denmark international, a proposal Ajax were reported to have laughed out of the room.
Monday evening brought the fresh attempt from the Frankfurt hierarchy, who are hopeful the improved terms can bring an agreement closer. Dutch reports suggest Ajax value the right-back at somewhere between €7.5m and €8.5m, so a gap remains between the parties.
Should the clubs settle on a fee, the remaining pieces would fall into place quickly. Gaaei has agreed personal terms with Frankfurt and is in line for a contract running until the summer of 2031.
The 23-year-old joined Ajax from Viborg in 2023 for €4.5m and has gone on to make 111 appearances for the Dutch giants, chipping in with seven goals and 13 assists from full-back. His current deal in Amsterdam runs until 2028, which strengthens Ajax's hand at the negotiating table.
In Frankfurt, Gaaei has been earmarked to fill the gap left by compatriot Rasmus Kristensen, who departed for Midtjylland in a €6m deal earlier in the window. The Eagles are rebuilding under the returning Adi Hutter, who has been handed the task of restoring European football to Deutsche Bank Park after the club missed out last season.
The Dane, who has a senior international cap to his name, was present at Ajax training as normal on Monday, and there has been no suggestion of him downing tools to force the issue.
Ajax, for their part, are in no rush. Boss Michel considers Gaaei a useful member of his squad and would happily keep him for the coming campaign, leaving Eintracht Frankfurt to decide whether they are willing to dig deeper to land their man.