Streak, who lost his job at the weekend after complaining about the make-up of the national selection panel, was accompanied by his lawyer Chris Venturas and senior players Grant Flower and Andy Blignaut.
Chingoka was flanked by the ZCU chief executive Vincent Hogg and several members of the board of directors.
Senior Zimbabwe players met at a secret Harare location to decide their joint reaction to the ousting of Streak.
A Test player who asked to not to be named said there was "a hint of a possible strike, which is still not impossible".
The board say they have accepted his resignation but Streak's father Dennis, acting as his spokesman, is adamant he did not resign.
Streak had wanted two selectors to step down and said he would consider his position if this was refused.
Streak said he would attempt at Thursday's meeting to resolve the situation "for the future of the game in Zimbabwe and to map out the way forward".
He said he was unable to make any further comment. The terms of his contract do not allow him to discuss publicly any aspect of his employment.
Streak was expected to have to withdraw his complaints about two selectors in return for being allowed to continue as a player, with Tatenda Taibu taking over the captaincy - as has already been announced.
A resolution needs to be found quickly. Sri Lanka are due in Zimbabwe for a four-week tour of two Tests and five One-day internationals on April 17.
If the dispute cannot be settled before next Wednesday the tour might have to be called off.