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Elliott made 18 league appearances under Arne Slot last season, but his only two starts came against Chelsea and Brighton, when Liverpool had already won the league.
That was in stark contrast to the previous season, with 11 starts in 34 league games for the Reds in Jurgen Klopp's final campaign.
Elliott has made 147 appearances, scoring 15 goals, in his six years at Liverpool since joining from Fulham as a teenager, with a season on loan at Blackburn in 2020-21.
England Under-21 boss Carsley has only seen Elliott's desire to succeed this summer.
"I've not seen a lot of the frustration," he added.
"We're very lucky to have him."
Brighton have been linked with a move for Elliott, as have former club Fulham, and while Wolves have been mentioned they have already signed Spaniard Fer Lopez and any fee is likely to be too high for the Molineux outfit.
Former Liverpool defender Stephen Warnock, part of BBC 5 Live's commentary team in Slovakia, believes Elliott has done well to ignore talk about his future and impress at the Euros.
"There's a lot of speculation about 'will he be at Liverpool next year?'," said Warnock, who made 67 appearances after coming through the youth ranks at Anfield.
"Because of the amount of games he has played for Liverpool, and the impact he had coming in, I think we all thought he would catapult and play for a long time in the first team.
"But he has a World Cup winner in Alexis Mac Allister in front of him, [Ryan] Gravenberch had an unbelievable season and [Dominik] Szoboszlai was brought in for big money.
"He has responded well in this tournament.
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