Borussia Dortmund are demanding £108m for the England international and the deal appears to have stalled, with United unwilling to match that number.
The financial effects of the ongoing coronavirus pandemic means that no other clubs have seriously explored the possibility of signing Sancho and Carragher believes Liverpool will enter the race next summer if United do not pay the money and sign Sancho now.
Speaking to the Sunday Mirror, Carragher said: "United have got a clear run at Sancho now – but that will change if he’s still at Dortmund next year.
"I’d be very surprised if Liverpool didn’t try to sign him next summer. That’s why I think United will pay what they have got to pay.
"When United have paid big fees for players in the past we’ve all laughed and said, 'they’ve been done again'. People called Woodward a joke – selling clubs could always get an extra £10m or £20m from him.
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"Now Borussia Dortmund have asked £108m for Sancho. And because United haven’t paid it straight away everyone is criticising them.
"Woodward is damned if he does and damned if he doesn’t."
Carragher also claimed that the addition of Sancho could turn United into title challengers, allowing them to compete with Manchester City and Liverpool.
"I don’t think that Manchester United are as far away as people think," he added. "One or two more signings could make that difference.
"For me, signing Bruno Fernandes in January completely changed the club. He came in and suddenly United were a completely different team. That’s how close it is. It could be that Sancho comes in and suddenly United are challenging for the title.
"But unless United get a couple more in then I can’t see how they will challenge yet. You need 95 points now – and I think only City and Liverpool can get that."