Not in Florida, and not likely in New York either.  Lender retained attorney to start a foreclosure.  He did not.  Lender started legal malpractice case, and assigned both the foreclosure, the loan, the mortgage, as well as the legal malpractice case.  Florida court did not permit the suit,  Hinshaw reports:

"Florida Supreme Court underscores adherence to not permitting assignment of legal malpractice claims.  Law Office of David J. Stern, P.A. v. Security National Servicing Corp.,___So. 2d___2007 WL 1932251 (Fla. 2007)

In Security National Servicing Corp. v. Law Office of David J. Stern, P.A., 916 So.2d 934 (Fla. App. 2005), the District Court of Appeal of Florida, Fourth District, cited prior Florida precedent regarding cases in which a legal malpractice claim arose from a lawyer’s failure to timely file a foreclosure action, such as that before the court. An appeal to the court followed. While the appeal was pending, the client assigned both the subject loan and the client’s legal malpractice claim to plaintiff.

The trial court granted summary judgment in favor of defendant on the ground that the cause of action was not assignable. The appellate court acknowledged that rule, but distinguished the case on appeal, stating that the subject policy factors were not present. Specifically, plaintiff was not an adversary; there was no risk of disclosure of confidential information; and there was no risk of the commercialization of legal malpractice claims. Citing Cowan Liebowitz & Latman, P.C. v. Kaplan, 902 So. 2d 755 (Fla. 2005) and Cerberus Partners, L.P. v. Gadsby & Hannah, 728 A.2d 1057, 1061 (R.I. 1999), the court allowed the assignment."