Speed was having talks in the afternoon with the ZCU chairman Peter Chingoka and chief executive Vincent Hogg immediately following a 45 minute session with three senior players and one of the more junior ones, who had been nominated at an earlier player meeting.
One of them, who did not wish to be identified, said there had been a "most productive discussion with Mr Speed and we were very pleased to see him in this way.
"He now has a very clear understanding of our position. But I cannot give you any details about what was said on either side."
Speed went on to have his first formal meeting with the ZCU since arriving more than two days ago from London. His only contact up until Tuesday was a conversation with Hogg, which the senior ZCU official confirmed.
The ICC chief executive was not immediately available for comment.
The player said he and his colleagues don't hold out much hope for a breakthrough at this stage.
"We're not 100 percent sure he can actually do anything. We don't know how far he is prepared to go over this issue. It's our impression that Mr Speed is none too happy with the ZCU for not meeting him formally until today (Tuesday).
"I can say we are still prepared to mediate, as we have been all along. We have done our best to find an answer."
The impasse began seven weeks ago when former captain Heath Streak demanded changes to the national selection panel and was fired. The players went on strike demanding his reinstatement.
The 12 "dissident" players decided at their meeting to resume practice - though informally - which means not under the direction of official coach Geoff Marsh. "And we will not be putting ourselves forward to play against Australia" the player said.
Rumours were circulating among Matatebeleland cricket officials in Bulawayo on Monday that the ZCU, Speed and Australia's management were holding talks about possible changes to the forthcoming Australia tour. But Hogg denied that any talks had taken place.