The fourth-ranked Murray has been looking for a second coach to work alongside Amelie Mauresmo since splitting from long-time team member Dani Vallverdu in November.
The 42-year-old Bjorkman will spend a week with Murray some time next month ahead of potentially joining the player's coaching team on a permanent basis. Bjorkman is currently on a Swedish reality television show.
Murray said last week that his poor recent performances in tournaments in Rotterdam and Dubai, where he was eliminated in the quarterfinals, were because Mauresmo was occupied with Fed Cup coaching duties with her native France. Mauresmo is only committed to Murray for 25 weeks per year.
Bjorkman retired in 2008 after winning nine Grand Slam doubles titles and reaching a career-high singles ranking of No. 4 in 1997.
Murray is playing in Indian Wells this week.