ACA has increased the pressure on CA to lift the ban on Smith and Warner ever since an independent review claimed that the "culture" nurtured by the board contributed to the ball-tampering scandal, which rocked the nation in March.
Smith and vice-captain Warner were each suspended for 12 months after the ball-tampering scandal broke in March, while opening batsman Bancroft was banned for nine months.
CEO, Kevin Roberts admitted that the governing body is considering to lift the ban on the trio after repeated appeals from the players union. "They wrote to the board rather than to me or all management".
'So not for me to comment on a board matter other than to say that the board will be respecting that submission and giving it due consideration'.
The bans extend to the domestic Sheffield Shield and the Big Bash Twenty20 competition.
Earlier on Wednesday, Cricket Australia booted out high-performance chief Pat Howard as well as broadcasting boss Ben Amarfio in a bid to purge their image after the Longstaff review.
The one-day team, now led by Aaron Finch, have lost their last seven including a 5-0 whitewash by England and a stinging defeat in Sunday's series-opener against South Africa in Perth after Pakistan had completed a 3-0 sweep of the Twwenty20 worldwide series in the UAE.
The arrival of Warner, Bancroft, and Smith will strengthen their Test squad as they are gearing up to take on India in a four-match Test series that gets underway in Adelaide on December 4.
Roberts, who was also part of the tumultuous pay dispute with the players past year, wants to fix the relationship with the players and that it remains a top priority.
And Roberts added: 'I think what's really important is that we at Cricket Australia need to embrace the players and we need to make sure that the players feel a really valued part of CA. "There's no doubt that we need to face up to the review and the fact that it acknowledged that the organization hasn't been united enough across the playing and long-playing sides", Roberts said.