The Portugal international defender is in Turkey to finalise the move and the Dogan press agency reports that an agreement is close.
The 30-year-old won the UEFA Champions League with Porto and Chelsea but endured a troubled first season with QPR, which ended in relegation from the Premier League.
The full-back was fined two weeks' wages for refusing to sit on the bench against Fulham in December and was taken off at half-time during a late-season game against Newcastle after being booed relentlessly by the QPR support.
He now looks set for a switch to a side who have won the Turkish league six times but only finished ninth last season, although they did reach the Turkish Cup final and qualified for the Europa League.
Bosingwa was greeted at the airport in Turkey by his former Chelsea team-mate Florent Malouda, who recently joined Trabzonspor himself.