Gibson quit after Euro 2012, when he didn't feature in any of Ireland's three group matches.
Trapattoni has since tried to contact the Everton midfielder, but despite receiving no response, he insists that doesn't mean he is not welcome back in the international fold.
"It is very difficult to know what Darron's position is when he does not answer his text messages," Trapattoni told The Daily Star.
"We have also tried to contact him through his club, but it is no problem.
"I haven't given up on him. I still wait for a player like Darron Gibson. The door is open."