The Independent says an offer of £17 million was rejected by the Dutch side, who continue to insist they will not sell for anything less than 25m.
Suarez was in the stands in Amsterdam for Ajax's 4-1 win in a Dutch cup tie with NAC Breda. There had been some suggestions that he might not turn up, in protest at the club's refusal to negotiate down on price.
But Liverpool appear now to have only 24 hours to match the asking price or proceed through the season without the 25-year-old.
The Ajax board have set a deadline of tomorrow for the sale to take place.