Ozil, 29, announced his international retirement last month after Germany crashed out of the World Cup. He said he had faced "racism and disrespect" because of his Turkish roots.
Ozil, who had been a key member of the 2014 World Cup-winning side, had faced a barrage of criticism for having his photograph taken with Turkish President Tayyip Erdogan in May and for his performances at the World Cup in Russia.
In his resignation statement last month, Ozil, who has Turkish roots, also accused the DFB and Grindel of failing to back him in the row over the photo.
In an interview with Bild newspaper on Sunday, Grindel said he should have handled things differently.
"I should have positioned myself more clearly given the racist attacks from some corners and I should have put myself in front of Mesut Ozil," Grindel said.
"Such attacks are unacceptable. I should have found clear words."
Ozil had been jeered in Germany's last warm-up games before the World Cup and many German fans had called for him to be dropped from the squad over the photo.
"I am sorry he feels let down by the DFB," Grindel added.
Ozil was made one of the scapegoats for Germany's earliest exit at a World Cup for 80 years in Russia, the side failing to get out of the group stage despite being the defending champions.
Grindel, however, said he had never blamed Ozil for that failure.
"It was always clear for me that we win together and we lose together. It would be absurd to blame one single player for our elimination," the 56-year-old, who took over the DFB in 2016, said.
He also ruled out resigning over the affair, saying he felt "strong backing from regional federations and the Bundesliga".
Grindel said there had been no discussion between Ozil and national coach Joachim Loew and that any possible return to the team would need to be discussed among the board and the coach.
The DFB president also said he would work towards bridging the gap that has opened up in recent months between fans and the national team.
Some of the options could include scheduling more open training sessions and lowering ticket prices, while the overall marketing concept of the team would also be reviewed.
"We want to come close again with the fans," he said.
(Reporting by Karolos Grohmann; Editing by Peter Rutherford)
