Postecoglou lasted just 39 days in the job after replacing Nuno Espirito Santo, who was sacked just three matches into the 2025-26 campaign.
Dyche took over at the City Ground with Forest 18th in the table after one win in eight games during a difficult start to the season.
After a 2-0 defeat by Bournemouth in his first match in charge, Dyche led Forest to four wins and one draw in their next seven to guide them out of the drop zone.
A run of four consecutive defeats followed, but Dyche’s side responded with a four-match unbeaten run – including a 2-1 win at fellow strugglers West Ham and a resilient goalless draw with leaders Arsenal.
However, the improved form of Leeds United and West Ham has kept Forest mired in the relegation battle.
Forest finished 13th in the Europa League group table, with four wins, two draws and two defeats in their eight matches, to reach the knockout phase.
They face Turkish side Fenerbahce over two legs this month for the chance to reach the last 16, with the first leg taking place next Thursday.
Forest are out of both domestic cup competitions, losing 3-2 to Swansea under Postecoglou in the EFL Cup third round in September and exiting the FA Cup on penalties to another Championship side, Wrexham, in the third round under Dyche last month.