Van Persie fell out of favour with Van Gaal following a below-par season, in which he scored just 10 goals in 27 Premier League appearances, and as a result the 31-year-old striker decided to sign with Fenerbahce last week for £4.7 million.
Van Persie admitted that alarm bells started ringing when he was an unused substitute during United's 1-0 loss to Chelsea on April 18.
"That was one of the first signals things weren't going in the right direction," he told the Sunday Times. "I asked to play in the reserves, to get my minutes, but after was on the bench again.
"The atmosphere changed between me and Louis and people at the club saw it, but I was always professional. At that point I didn't think to leave. [My wife] Bouchra was happy. The kids were happy. I was happy in Manchester."
Van Persie then received confirmation that his time was up when Van Gaal told him that he was not in the club's plans for the coming season.
Recalling his conversation with Van Gaal at the Red Devils' annual charity golf day, Van Persie said: "I was still thinking we could come back from holiday and start from scratch. He had changed his mind about me before. When he took over Holland he said to me 'You're the No. 3 striker.' I was 'O…K…' but I fought and became the No. 1 and his captain.
"But when I came back, it wasn't an honest battle any more. Fighting to get back in the team wasn't given [to] me as an option. He was sending me to Pitch Two [reserves].
"And I'm a mature player. I'm not stupid. I didn't get angry or emotional. These things are part of football, part of life. You have to make the best out of any situation so I'm doing this by moving on."
The Netherlands international enjoyed a superb debut season with United after signing from Arsenal for £24 million in 2012, scoring 26 goals in 38 Premier League appearances. But he struggled to replicate that form in the two seasons that followed, and dropped down the pecking order as a result.