Allardyce took charge of the Toffees after Rooney had returned to Everton from Manchester United for his second spell with the club.
Allardyce told the Tippy Tappy podcast: “Wayne is one of the best players this country has ever seen in goalscorers, but my problem with him was that the fact he was coming to an end, and he was back at the club he loved, which is Everton.
“Sometimes I was really worried about bringing him off when he was looking tired, but I thought he understood it quite well. For England, it was all about him not playing.
“With the press, I got a first-hand view of how much pressure he’d been under all his life, by being a world-class player.
“The hardest thing as a manager is telling somebody like him that you think they’re struggling to live up to that level of football, because you won’t accept it, and I didn’t. But you have to make that decision.”
Allardyce added: “Wayne Rooney at Everton was a leader, and I regret not making him captain straight away.
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"I should have just said to the club: ‘This is Wayne, he loves the club, he’s loved it since he was a kid, so he’s going to be captain.’”