Chairman Peter Chingoka said on Friday that the ZCU were ready to engage in a "dispute resolution tribunal" to resolve the problem with 15 white players who went on strike after Streak's sacking.
"We have been working on this for six weeks. We are suggesting a chairman who would be agreed by both parties and one person appointed by each side," explained Chingoka.
The offer was discussed on Wednesday by Alwyn Pichanick, lawyer for the ZCU and Chris Venturas, legal representative of the players.
Venturas explained: "It amounts to the same thing as a tribunal, for its decision will be legally binding and it will examine all the factors of the dispute together with everything that led to it.
"This has been a long drawn out, stressful, volatile and unproductive business. Consequently many of the players have created new lives for themselves. I don't know their reaction yet," he added.
Meanwhile the International Cricket Council (ICC) dismissed reports that they have given the ZCU a 14-day deadline beginning on June 30 within which to resolve the dispute.
"What the ICC has announced is that it has given the parties 14 days within which to agree a mechanism or process to solve the dispute, not to resolve the dispute itself," ICC president Ehsan Mani said.
"We recognise that this is a Zimbabwean dispute and our clear preference is to have it resolved in Zimbabwe by Zimbabweans," said Mani.
"If there is no agreement on the process, the newly elected ICC vice-president, Percy Sonn, and I will make a final decision on application of the ICC's disputes resolution process after this 14-day period.
"The ZCU is firmly of the view that this system has no jurisdiction but the ICC's legal advice is clear in saying that it does."
The ZCU had for two months resisted the players' call for independent arbitration - or a tribunal. In their turn the players refused the Union's preference for internal mediation.
Seven former Zimbabwe players left Harare on Friday to join seven others from different parts of the world for a three-weeks charity cricket tour of England. They call themselves the Red Lions.
The party of 14 - fast bowler Andy Blignaut is not available - will be joined there by Venturas on Monday.
They will formally discuss the offer and decide whether to accept it. Afterwards they are scheduled to talk to ICC chief executive Malcolm Speed.
The Zimbabwe cricket crisis began on April 2 when Streak accused the national selection panel of making some team choices based on race rather than merit. He was immediately and angrily dismissed.
The other players, all white, having failed to get him reinstated, went on strike. They in turn were sacked and their contracts revoked.
A young and inexperienced Zimbabwe squad took over, but after a series of heavy defeats by Sri Lanka the ZCU was forced to accept a six months period in which they do not play Test cricket, causing considerable financial losses because of unplayed tests scheduled against Australia, Pakistan and England.
They are due to resume Tests in Bangladesh during January, following intensive coaching and two incoming 'A' tours by India and South Africa to provide experience, after which their status will be reviewed.