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Shakespeare was appointed Foxes manager until the end of the season on Sunday, having been caretaker boss since Claudio Ranieri was sacked in February.
Leicester have won both of their Premier League games under Shakespeare.
"On behalf of all the players we'd love to see him get the job long term," said Foxes midfielder Danny Drinkwater.
"We were all very happy to see him get it short term - I'm sure it would be the same reaction longer term."
Shakespeare said the club would "have plenty of time in the summer to take stock and evaluate".
Leicester host Sevilla in the second leg of their Champions League last-16 tie on Tuesday, trailing 2-1 from the first leg but with an away goal thanks to Jamie Vardy's strike in Spain.
Asked whether Premier League champions Leicester could win the Champions League, Shakespeare said: "Why not? We're in it and we have to try to be competitive every game."
He added: "'I'm expecting the team to take the momentum from the last two results and really play off the front foot."
Shakespeare, 53, was Ranieri's assistant last season after being brought to the club by the Italian's predecessor, Nigel Pearson.
He confirmed he had not spoken to Ranieri since his appointment was confirmed on Sunday but said there were "no grudges" between the two.
"I bear no grudges, Claudio bears no grudges. We have to try and move on as a football club," said Shakespeare.
While caretaker boss, Shakespeare was backed by 92% of readers in a local newspaper poll to take over full time, following wins against Liverpool and Hull City.
However, Shakespeare added: "If we'd have lost both games, it would probably have been 1-2% and that would probably have been my family."
New Leicester boss Craig Shakespeare says he is "not looking beyond this season" despite being backed by his players to get the job longer term.