Veh, who led VfB Stuttgart to the league title in 2007, has rejected reports he had a specific offer from Schalke 04, but has yet to sign a contract extension with his current club, demanding a more competitive squad in order to stay on.
"I am close to Armin and openly discuss with him the current situation," said Eintracht sports director Bruno Huebner on Saturday.
"What I can say is that we remain his first choice for any talks and the possibility of holding on to him is great. I am confident," he told reporters.
Veh said any discussion would follow Sunday's game against Stuttgart.
"If we win this game or at least get a point then it will be relatively quick," he told reporters. "Then there will be a decision within the next 10 days. I am not one to drag things out."
With a point Frankfurt would have achieved their aim of 40 points for the season with eight games left.
Frankfurt's run this season has been nothing short of sensational with the promoted team currently in fifth place and challenging for a Europa League or even a Champions League spot.
"I have threatened Armin that I will drown his dog in the river Main if he does not stay," joked Frankfurt boss Heribert Bruchhagen on Friday.