Brian McDermott was last week relieved of his duties at the Madejski Stadium helm after a run of four successive league defeats left the Royals in serious danger of an immediate return to the Championship.
Reading's Russian owner Anton Zingarevich is believed to be in the country as the search for a replacement continues, with Poyet the latest favourite for the job.
After links to former Swindon boss Paolo di Canio and ex-Southampton manager Nigel Adkins, the odds on the Brighton boss replacing McDermott have shortened to as low as 1/10 with some bookmakers.
It is understood tentative talks have taken place regarding a move to Reading but that it is by no means a done deal, with Poyet's Brighton side upwardly mobile.
They are seventh in the Championship after a 3-0 derby defeat of Crystal Palace and Poyet has previously stated he has ambitions to go to one of England's elite.
However, should the Uruguayan plump for a move to the Royals, skipper McAnuff believes he would be joining a club that could still beat the drop.
"No one's giving up," the midfielder told the club's official website.
"We're got eight massive games left now and it doesn't matter who we've got, we'll show our pride and see if we can pick some results up.
"For us it's about believing that we're good enough to be here. We really do believe that - we're here on merit.
"We've earned the right to be here and we just want to make sure we give it a good crack between now and the end of the season and do our best to make sure that we stay up.
"Credit to Eamonn [Dolan, caretaker manager], the staff and the players - it was a tough week but in football you've got to roll with the punches and get on with it, and I thought we did that on Saturday."
That match saw Reading fall to champions elect Manchester United, with Wayne Rooney's first-half goal proving the difference.
The Royals remain second from bottom and seven points adrift of safety, although McAnuff was proud of the performance his team-mates put in at Old Trafford.
"Obviously lots happened in the week and it was nice to get out there and do what we're here for and play football," he said.
"As usual, it seemed as if the game was won before we had even got there.
"There were some pretty ridiculous predictions before the game but we came up and once again showed what we're about."
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