The Baggies look set to land the 26-year-old Sweden international after Swansea opted not to pursue their interest.
The Swans' interest in Olsson had looked set to free up their left-back, Neil Taylor, to move to the Hawthorns.
But the Welsh club have decided to stick with Taylor and the Baggies have moved for Olsson, with Dorrans moving the other way.
The clubs are still negotiating terms and a cash fee might also be involved, but both moves appear likely to be completed.
Albion tried to sign Olsson two years ago but Norwich eventually won that race.
He has continued to impress since their relegation to the Championship and Tony Pulis now looks set to land him, in a move that would free up Joleon Lescott to return to centre-back.
Sebastien Pocognoli started the season at left-back and Chris Baird has also filled in, but Pulis has preferred Lescott there in Albion's last two games.
The former England centre-back admitted at the weekend that he was keen to revert to a centre-half role.