Rosen left Weil in July this year and has been in talks with several rival firms since the beginning of August.
The hire would be a particular coup for DLA Piper, helping it expand out of its mid-market niche.
A source close to DLA Piper said: "I know he's been talking with a number of other firms, but it's my understanding that he's very likely to go to DLA."
Having joined Weil when the firm launched in London, Rosen has built a strong reputation in private equity, advising on a number of high-profile deals. Clients include Blackstone Group, for which Rosen acted when it acquired Cadbury Schweppes' European drinks business for £1.27bn in 2005.
Rosen's exit left Weil with five private equity partners, including Marco Compagnoni, who joined from Lovells in 2006.
DLA Piper has 31 private equity partners across eight offices in the UK, 10 of whom are based in London. The firm declined to comment.