Yedlin, 27, is to travel to Turkey for his medical after both clubs agreed on a fee for the right-back.
Yedlin's contract was up in June and his original preference was to stay at Newcastle, but he has taken up a three-and-a-half-year deal with Galatasaray.
The two clubs were locked in negotiations for much of last week. Galatasaray tabled the contract offer to Yedlin on Tuesday, leaving it up to the two clubs to agree a deal.
Newcastle approved Galatasaray's latest approach on Saturday and gave Yedlin permission to travel to Istanbul.
A source close to the deal said Newcastle gave the U.S. right-back assurances he would feature in the reminder of the season but with no new contract offer on the table, the club allowed Yedlin to pursue a permanent deal elsewhere.
Once Yedlin has passed the medical, he will join Galatasaray on a deal based on future fees around performance and appearances.
Yedlin should now help bolster Galatasaray's push for the Super Lig, with the club sitting third in the Turkish top flight.
He is hugely experienced, with 62 caps for the U.S., and will leave the Premier League having played for both Tottenham and then Newcastle.
He joined Newcastle in 2016, and made 125 appearances for the club. He broke into Steve Bruce's starting lineup in December but his progress was then hampered by visa issues that have since been solved.