Klaas-Jan Huntelaar was Schalke's second choice target to strengthen their attack in the January transfer market.
Bild in Germany has reported that Schalke are eager to bring the 37-year-old striker back to the club to help boost their hopes of survival in the Bundesliga this season.
The Gelsenkirchen side are second bottom of the table with seven points from 15 matches, having picked up their first win last Saturday when they hammered four past Hoffenheim at home. Only Arminia Bielefeld, who are 15th, have scored fewer goals than Schalke.
Huntelaar was the second most prolific scorer in the German side's history, having scored 126 goals in his seven-year spell there before he moved back to Ajax in the summer of 2017.
The Netherlands international has announced he will retire from professional football at the end of the season, but when asked by Fox Sports if he could turn back on that decision, he said: "Never say never".
Schalke are hoping he can be convinced to return to the Bundesliga before he calls time on his career and it has been reported that sports director Jochen Schneider has already travelled to Amsterdam to discuss a possible deal. It has also been claimed that Huntelaar is open to moving across the border once again.
But Goal and SPOX understand that they made another option their main priority, with 36-year-old Vedad Ibisevic their favoured candidate, even though they just released him weeks ago.
Ibisevic joined Schalke from Hertha Berlin last September on a one-year contract, but announced the termination of his deal just three months later. He made just four top-flight appearances for Schalke this season, starting one.
"Vedad and we had different expectations of his involvement with Schalke 04. From our point of view, it is now better to end the cooperation and to part on good terms," Schneider said at the time, insisting that an argument with coach Manuel Baum, who was himself fired in mid-December.
Nevertheless, the director reached out to the Bosnia-Herzegovina international once more, only to be turned away.
Schneider turned to another former Schalke star in their desperate search for an experienced striker.
Huntelaar still has close ties to the club, telling Schalke TV in the summer that he has kept in contact with many members of staff, even meeting with kit managers and physios over coffee and cake.
The former Real Madrid and AC Milan striker has featured 11 times in the Eredivisie for Ajax and scored seven goals.
He made an immediate and decisive impact for Erik ten Hag's team on Thursday when he came off the bench in the 89th minute against Twente.
The two sides were tied at 1-1, but Huntelaar went on to net two goals in the dying stages to seal the three points as Ajax moved three points ahead of Feyenoord at the top of the table.