The Scot has been looking for a second coach to work alongside Amelie Mauresmo since splitting from long-time team member Dani Vallverdu in November.
Swede Bjorkman will spend a week with Murray at some point in the next month with a view to joining the team on a permanent basis.
Bjorkman, 42, was among the names touted as a potential replacement for Ivan Lendl when his successful coaching relationship with Murray ended a year ago before the Scot eventually plumped for Mauresmo.
A popular and well-respected figure within tennis, Bjorkman has remained in the game since retiring in 2008, working behind the scenes at the ATP tournaments in Bastad and Stockholm and in the media.
The exact timing of his week with Murray is up in the air because Bjorkman is currently appearing on the Swedish version of Strictly Come Dancing.
Murray is linking up again with Mauresmo in Indian Wells this week for the first time since the final of the Australian Open, with the Frenchwoman’s commitment to the partnership only for 25 weeks a year.
That is more than Lendl spent travelling with Murray, and having Vallverdu alongside him during the other weeks was of great importance to the 27-year-old.
Murray suffered poor losses to Gilles Simon and Borna Coric at tournaments in Rotterdam and Dubai last month and admitted he struggled without the support of a coach.
Speaking prior to helping Great Britain defeat the United States in Davis Cup last weekend, Murray said: “After the Aussie Open I spent the next three or four weeks with no coach and I feel that’s something I obviously need to get sorted so that when I get to the clay-court season I am not in that position, because I feel like there are some things I need to work on all the time and when I don’t have someone there it’s harder to do that.”
Bjorkman attained his highest ranking of number four in 1997, the same year he reached the semi-finals of the US Open.
He also made the last four at Wimbledon nine years later and was an exceptional doubles player, winning nine grand slam titles.
Murray and Bjorkman faced each other once in singles, with the Scot triumphing in five sets in the second round of the US Open in 2007.
Murray, who climbed back to world number four thanks to his Davis Cup victories over Donald Young and John Isner, will begin his Indian Wells campaign later this week against either Vasek Pospisil or Mikhail Kukushkin.
Bjorkman revealed his excitement about the opportunity, writing on Instagram: “I’m super excited and I’m really looking forward to join @andymurray and his team for a test week, which we both hope will lead to joining the team permanently! #teammurray.”
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