Sycamore Partners is nearing an acquisition of Walgreens Boots Alliance Inc., people with knowledge of the matter said, in a deal that could end the drugstore operator’s tumultuous run as a public company.

The companies are putting the final touches on a transaction that may be announced as soon as this week, sources said.

Walgreens Boots Alliance is closing in on a deal with private-equity firm Sycamore Partners that would take the struggling drugstore chain off the public market for around $10 billion, according to people familiar with the matter.