Renner takes over the Bourne franchise from Matt Damon in the latest film, playing Operation Outcome agent-turned-fugitive Aaron Cross.
Filmmaker Gilroy has now admitted that Renner faced tremendous pressure in starring in the first Bourne film without Damon.
"The bar was so high, Matt Damon had set the bar so high with Jason Bourne," Gilroy said, reports Flicks and Bits.
"And there's such an expectation with these films, they're not just pictures, they really have a rich emotional life and the human behaviour is real, you feel like it's really happening."
Gilroy went on to insist that he was ultimately very happy with Renner's portrayal of Cross.
"We had the opportunity with Aaron Cross, somebody who's just as soulful and just as complicated, so we really needed an amazing actor - and Jeremy Renner is that," he explained.
"We also needed someone who is physically extraordinary, because we knew what we wanted to do and we knew what was expected."
He continued: "Then we also needed, as the third piece, which was perhaps the most difficult, we needed someone who has those two skills, those two attributes, but wasn't so clearly identified in the audience's mind - that they didn't have a perfectly clear picture about them as another character, or an idea about them.
"We wanted to identify him as our guy, who was just enough out of focus to bring him forward. That was the trifecta that we were looking for."
Gilroy previously stressed that he played no role in Damon leaving the Bourne film series, adding that he also has no idea about plans for another movie.
The Bourne Legacy is due for release on August 10 in the US and August 13 in the UK.
Watch a trailer for The Bourne Legacy below: