The Titans have placed Cobb on injured reserve/recall with a calf injury. He can return to practice after six weeks and to games after eight weeks.
West was a third-round pick of the Browns in the 2014 draft. Sankey was the first running back chosen that year. West was the sixth.
Last year, as the Titans sorted through running backs, they had West in for a visit.
He led the Browns with 673 yards rushing and scored four touchdowns in his rookie season.
The trade leaves the Browns with two running backs on the 53-man roster: Isaiah Crowell and rookie Duke Johnson, who has been sidelined since he was diagnosed with a concussion in the third preseason game Aug. 29.
ESPN Browns reporter Pat McManamon contributed to this report.