Liverpool manager Rafael Benitez says they should be handed a place in next season's Champions League if they win the competition this season.
Benitez's side are fifth in the Premiership
The Reds could deny rivals Everton, who are favourites to finish fourth in the Premiership, if they miss the top four but are crowned European champions.
Liverpool would need the Football Association's approval to replace them.
But Benitez said: "Common sense says that if you win a trophy next year you need to defend this trophy."
But Everton chief executive Keith Wyness said: "It's also common sense that the domestic league, which is 38 games as opposed to a cup competition which probably is about 16 or 17 games, should also take priority.
"I understand the thought that the cup winners want to defend their trophy but nevertheless I do believe that the domestic league is the cornerstone of football."
Ex-Everton boss Howard Kendall also thinks they deserve Champions League football if they finish fourth in the Premiership.
Kendall, whose 1985 side were excluded from Europe after the Heysel disaster, told BBC Five Live: "Evertonians will be looking back and thinking 'oh no, not again'.
"We believed we were the best then and we were denied the chance to prove it."
Kendall's team won the league in 1985, the year in which English clubs were banned from Europe after Heysel.
"We won the league that year, the Cup Winners' Cup and we were looking forward to the European Cup," he said.
Of the current side, Kendall added: "Everyone was tipping us for relegation with the sale of Wayne Rooney. I think they've done a magnificent job and deserve to go into the Champions League."
Everton will be back in Europe for the first time in 10 years following Tottenham's defeat at the hands of Arsenal.
It means they cannot finish below seventh in the Premiership - guaranteeing them at least a Uefa Cup place.
Uefa reaffirmed on Monday it will not offer an extra Champions League place for an English team next season even if Liverpool win this year's tournament and finish outside the Premiership's top four.
If that happens the FA will have to decide who enters next season's competition - the fourth-placed team or the Reds.
And Toffees boss David Moyes is fully focused on claiming fourth spot.
He said: "The players have done everything they can, we are really pleased with the way the season has gone here and we are optimistic that we can finish on an even more positive note.
"Considering what people were saying about us at the beginning of the season, this is a magnificent effort by the lads."