Everton have been on the lookout for a new manager since axing Ronald Koeman last month, with the Toffees currently enduring a nightmare Premier League campaign.
With just three wins from their first 11 league games, Everton currently sit just two points above the relegation zone.
Express Sport understands they are lining up Allardyce to replace Koeman, with David Unsworth currently in temporary charge.
But The Sun say Everton only want Allardyce until the end of the season, where they hope to make a splash by landing a bigger, more marquee name.
Should the former England boss take the reins, then former Leicester boss Shakespeare will be his No2.
The 54-year-old was axed by the Foxes last month, just four months after penning a three-year deal.
Allardyce has supposedly told Everton he wants a long-term contract.
But the former England manager could be tempted to take over at Goodison Park if a lucrative offer is put in front of him.
Meanwhile, Sky Sports’ Kaveh Solhekol claims Atletico Madrid boss Diego Simeone is Everton’s No1 choice to be their next permanent boss.
“Everton working hard behind the scenes to find a new manager,” Solhekol said.
“We understand they’re working on four options at the moment.
“The one that’s really, really interesting, the dream candidate we’re being told is Diego Simeone the Atletico Madrid manager.
“We understand he is the No 1 choice of Everton’s owner Farhad Moshiri, a man who has a lot of money. He’s the man who Moshiri wants.
“Of course it’s going to be very, very difficult to get Diego Simeone out of Atletico Madrid.
“A club who are unbeaten in La Liga, I think they’re fourth in the table. They’re not doing great in the Champions League at the moment, they’re in the same group as Chelsea and Roma.
“But he is Farhad Moshiri’s dream candidate. It would be very surprising to see him at Everton but Farhad Moshiri is a very rich man, he wants the best for Everton.
“He thinks that Diego Simeone would be the right man for the job.”