Scottish organiser Kevin Lindsay said more talks with the rail operator would take place on Monday.
He said the aim was to avoid industrial action by negotiating a deal that was acceptable to his members.
ScotRail has meanwhile introduced a temporary scaled-back timetable for Sunday services.
Mr Lindsay warned that the railways could be hit by a series of 24-hour strikes if the issue is not resolved.
ScotRail, which was nationalised by the Scottish government in April, is currently running a greatly-reduced timetable after many drivers refused to work overtime or on their rest days over the dispute.
But it did not put a Sunday timetable in place, resulting in services being delayed and cancelled at the last minute.
On Thursday, ScotRail announced to customers: “We’re introducing a temporary timetable for Sundays too, so we can provide greater certainty & reliability for our customers.
“It will be available to view on our website and app from Saturday morning.”
Details of the timetable on the ScotRail revealed there would be only one train an hour from Glasgow to Troon and Ayr. The route would normally be busy on a bank holiday weekend, with two services running.
Aslef has previously rejected a 2.2% pay offer, before later being offered 4.2% and improved conditions – which Transport Minister Jenny Gilruth said she was “hopeful” would be accepted.
But the union announced on Wednesday that its national executive had rejected the offer and would ballot for industrial action unless further talks were held.
ScotRail said it was frustrated by the decision, and “astonished” that the union had decided not to put the improved pay offer to its members.
It said the “substantially improved” offer reflected the cost-of-living challenges faced by families across the country, while balancing it against the need to provide value for the taxpayer.
Transport Scotland – the government’s transport agency – described the latest offer as being “both fair and affordable”.
When ScotRail made an improved pay offer to Aslef, there was optimism the package might be good enough to settle the row.
That was never certain but it was a surprise to see it rejected by the union at this stage.
ScotRail had described the 4.2% pay offer as its “final offer”. Aslef believe it may be possible to get a better deal.
Regardless of affordability, there is a problem.
ScotRail is now back in public ownership and covered by public sector pay policy.
The 4.2% offer compares to the 2% being offered to council workers and teachers.
Council workers are now being balloted on strike action.
What sort of pay rise will council and education unions consider acceptable if rail workers should get 4.2% or even more?
Speaking to the BBC’s Good Morning Scotland programme on Thursday, Mr Lindsay said the union had held a series of meetings with about 250 of its members – only one of whom said the offer was acceptable.
He added: “We will sit down with ScotRail again on Monday and we believe that we are close to reaching agreement. We think there’s a number of things that can be done to reach a settlement.
“We’re a trade union that has never been on strike in 20 years here in Scotland. We’re not even taking industrial action just now.”
‘Stressful and tiring job’
Mr Lindsay said drivers had to work an extra two days a month on average to keep the railways running, which he said showed that there was a “chronic shortage”.
He added: “Train driving is a stressful and tiring job and having quality time away from their workplace is something that we always encourage people to do, but we also support anyone who wishes to work additional shifts.”
He said the ball was now in ScotRail’s court to come back with an acceptable offer “using the money that’s already in this package” that would avoid industrial action.
If that did not happen, he said the union was likely to ballot its members for a withdrawal of all overtime working and a series of discontinuous 24-hour stoppages.
But he stressed: “We are a long way off from that. We’re a highly-trained team of negotiators and we’ll go in and we’ll speak again on Monday.
“Hopefully we can thrash out a deal that delivers for Scotland’s train drivers, but more importantly delivers for the people of Scotland as well.”
Mr Lindsay also dismissed suggestions that the union was being unrealistic in its pay demands at a time when other public sector workers such as teachers are being offered 2%.
He said: “I’m not going to apologise for train drivers making reasonable demands when we’re facing an earnings and income crisis.
“We cannot continually play one worker off against another.
“I am more than happy to support council workers and to support Cosla’s claim for the Scottish government to give more money to Scottish councils to ensure that Scottish council workers, teachers and everyone else gets a fair pay rises.”
ScotRail introduced an emergency timetable last month which it said would give customers a degree of certainty about rail services after the network was hit by hundreds of cancellations.
The timetable has almost 700 fewer services a day, with the last train on many routes leaving up to four hours earlier than before.
ScotRail managed to run extra services on Wednesday night to ensure fans were able to get to and from Scotland’s World Cup play-off match with Ukraine at Hampden.