The former assistant to Claudio Ranieri has taken over the reins at the Premier League champions following the dismissal of the veteran Italian manager in late February.
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Initially appointed as an interim coach, the 53-year-old led the relegation-threatened Foxes to consecutive wins in his first two games in charge, including an impressive 3-1 result over Liverpool.
“On behalf of all the players we’d love to see him get the job long term,” Drinkwater told BBC Football.
“We were all very happy to see him get it short term – I’m sure it would be the same reaction longer term.”
Leicester are currently 15th in the Premier League standings and face off against Sevilla at the King Power Stadium on Tuesday, for the second leg of their last-16 Champions League tie looking to overturn a 2-1 deficit.