Hart has won 63 caps for England, but had a disappointing Euro 2016, making mistakes against Wales and Iceland as the side was knocked out in the last 16.
With Hart out of favour with new City manager Pep Guardiola, Argentinian Willy Caballero has been an ever-present for the club so far this season.
Everton are keen on signing Hart, who is not attracted by the prospect of joining interested parties Sevilla and Borussia Dortmund.
Barcelona said on Saturday they had reached the “beginning of an agreement” with City over the transfer of their goalkeeper Claudio Bravo.
“We want as many players as possible in their club’s first team that join up with the squad, then go on to play for England and be successful,” said Allardyce.
“It doesn’t happen as much now as it did 20 or 25 years ago because there are a smaller number of [English] players who play in the Premier League now.”
England’s opening Group F qualifier for the 2018 World Cup will take place in Trnava on 4 September and will be their first game under Allardyce, who was appointed after Roy Hodgson resigned following the defeat to Iceland.
Allardyce is still to decide whether Manchester United forward Wayne Rooney will continue as captain.
“I will reveal the captain by Wednesday at the latest because everyone is asking the question,” he said.
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