The German car maker also said it would plead guilty to breaking certain US laws.
VW said it was in advanced discussions with the Department of Justice and US Customs about the deal.
The final agreement has yet to be approved by VW's management and supervisory board, which could happen later on Tuesday or Wednesday.
VW said it had negotiated a "concrete draft" of a settlement with US authorities that included criminal and civil fines totalling $4.3bn, as well as appointing an independent monitor for the next three years.
The $4.3bn fine means that the total costs associated with the emissions cheating scandal are set to exceed the $19.2bn the company has set aside to deal with the issue.
VW has already agreed to a $15bn civil settlement with environmental authorities and car owners in the US.
The scandal erupted in September 2015 when the US Environmental Protection Agency (EPA) found that many VW cars sold in America had a "defeat device" - or software - in diesel engines that could detect when they were being tested and adjust the performance accordingly to improve results.
The German car giant subsequently admitted cheating emissions tests in the US and many countries throughout the world, including the UK.
Arndt Ellinghorst, head of global automotive research at Evercore ISI, said the draft settlement was good news for VW and would "draw a line under all remaining US-related legal risk".
It was important that VW had "managed to come to an agreement that allows the company to move on", he said, adding: "It's a major relief that this doesn't get dragged into the new US administration."
Mr Ellinghorst expected the company to increase its provisions for diesel-related costs by up to €3bn to as much as €21bn.
On Monday it emerged that VW executives knew about emissions cheating two months before the scandal broke, but chose not to tell US regulators, according to court papers.
The bosses involved include Oliver Schmidt, who was in charge of VW's US environmental regulatory compliance office from 2012 until March 2015.
On Monday he was charged with conspiracy to defraud and has been remanded ahead of a court appearance on Thursday.
Volkswagen has agreed a draft $4.3bn (£3.5bn) settlement with US authorities over the emissions-rigging scandal.