Update, June 13: Boylen has accepted the offer, as expected, reports Adrian Wojnarowski.
The Spurs' assistant coaches are now Ettore Messina, Ime Udoka and Becky Hammon. It's unclear at this point whether Gregg Popovich will look for another assistant or make do with the ones already in place. Sean Marks returning to the bench after taking a front office job is probably a possibility as well.
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The Bulls asked for permission to talk to Boylen, which the Spurs granted and there have been discussions about Boylen leaving San Antonio to become associate head coach in Chicago. According to Wojnarowski the Spurs value Boylen and are interested in retaining him, but I'm afraid the offer could prove to be too tempting.
With Ettore Messina joining the Spurs' coaching staff last offseason Boylen ceased to be the lead assistant. A more prominent role in Chicago might be just what he needs to make the jump to head coach one day after being rumored to be in the running for some positions in the past. Nothing is definitive but it wouldn't be surprising to see him accept the job.
The Spurs would surely miss one of their top assistants, but with Ime Udoka and Messina on board and Becky Hammon with a year of experience under her belt, they should be fine even if they don't make any new additions.
Boylen has been part of Gregg Popovich's staff since 2013, when he was hired after the departures of long time assistants Mike Budenholzer and Brett Brown. He was head coach of the Utah Utes in the NCAA and was an assistant coach on the Bucks, Pacers, Warriors and Rockets before joining the Spurs.