BBC Sport says the two parties broke up in a positive fashion, with a final decision from the Gers board due this week.
Ancelotti is now forming his six-man backroom team as he prepares for his first move into senior management - whether that be at Ibrox or elsewhere.
Should he land the job, Ancelotti could bring with him departing Real Madrid midfielder Luka Modric.
The Croatia international, like Ancelotti, is leaving Real Madrid this summer.
Ancelotti was expected to be assistant coach to father Carlo with the Brazilian national team, but he is now remaining in Europe to take the next step in his career.
Carlo Ancelotti said at his presentation last week: "He is currently negotiating with a European team. So I thought it was right that he didn't come here. If he finds a team, I will wish him the best. Otherwise, he can come back to us at any time."