Barton never missed details.

Natasha knew Barton better than anyone, and she knew he wasn't as ignorant as he would have others believe. Barton was perceptive, and it was what made him the world's greatest marksman. He picked up on details none of them saw: patterns informing them where the enemy would attack, weaknesses the villain had forgot to fix, and he clued them in to those subtle shifts that gave the enemy away. Natasha wasn't there, but she knew for a fact that Barton had figured out the Tesseract quicker than any of the world-class astrophysicists S.H.I.E.L.D. had employed on the project.

Barton's eye for details, his ability to read people when nobody else could, is what brought her to S.H.I.E.L.D. instead of putting her in a casket six feet under. Natasha trusted her friend with her life….and sometimes his knowledge of her unsettled her. Times when she knew he suspected more about her than she wanted to divulge reignited an urge in her to leave. She couldn't hide from him like she could hide from the other Avengers. And after dumping all of her personal history onto the internet after HYDRA infiltrated S.H.I.E.D., Natasha knew she needed to create space between them until the Black Widow could cover up those details he would easily see. Because Cli-….Barton never missed a detail. Hawkeye could handle himself, so when he pointed out to her that Justin Hammer was attempting to escape, she grabbed Captain Rogers, not Barton, to follow him to his lair and "interrogate" him for a way to shut down his Adaptoids.