Generally, no. Washington and Oregon follow the "American rule," which is that each side pays its own attorney fees, unlike a "loser pays" rule. There are many exceptions to the American rule, in which the prevailing party can make the losing party pay the reasonably attorney fees it incurred - typically claims based on statutes like employment discrimination claims, for example. However, legal malpractice claims are not among these.
However, in Washington, and possibly in Oregon also, a plaintiff may avoid deducting from his or her damages in the legal malpractice case the attorney fees he or she would have had to pay the attorney in the underlying case. Often this can have the same effect on total recovery as if the plaintiff in the malpractice case could recover his or her attorney fees. See our other article - Do I have to subtract from my damages in the legal malpractice case the attorney fees I would have had to pay my first lawyer if we had won the case?
Also, attorney fees incurred in the legal malpractice case may be recovered if the fee agreement between lawyer and client so provides. However, this would require an unusually broad attorney fee provision. Normally, an attorney fee provision in a lawyer engagement agreement (if it has one at all) has something like the following language: "in any action to enforce this agreement, the prevailing party shall be entitled to recover its reasonable attorney fees." "Enforcing the agreement" will likely be interpreted as a claim for unpaid fees or costs, and the like. The obligation to pay fees arises from the contract. While it might seem counterintuitive, this is unlike a claim for legal malpractice, which is premised on common law duties of care and to avoid foreseeable harm, rather than the contract. So the fee provision cited above would not apply to a malpractice claim.
However, if the fee provision were broader, prevailing party fees might be available. For instance, suppose the fee agreement stated, "in any action arising out of this Agreement, the prevailing party shall be entitled to recover its reasonable attorney fees." This "arising out of" language is better for a plaintiff seeking recovery of fees, though the defense might still successfully argue that the reference to the "agreement" confines the provision to fee disputes. Suppose the fee provision went a step farther, and provided, "in any action arising out of this Agreement or the representation of Client by Attorney, the prevailing party shall be entitled to recover its reasonable attorney fees." Here we have a potential game changer and the claim for attorney fees would be strong. However, this scenario is rare.
A client should be careful what he or she wishes for, however. A prevailing party fee provision would probably be reciprocal, meaning that if the client loses his or her legal malpractice claim, he or she could be on the hook to pay the defendant's legal fees. This increase of downside risk might not be sufficient to compensate for the potentially higher recovery. It is true that if the claim for attorney fees is well-grounded, then the defense might ask for them anyway. However, it is possible that a plaintiff who does not plead and seek such fees will find that the defense follows suit.
Also, it is possible that an award of contract-based attorney fees against a malpractice defendant would be an uninsured liability , since the malpractice insurer would likely argue it is not responsible to indemnify its insured for a contractual obligation. That can be a two-edged sword in itself. On the one hand, the risk of personal exposure in excess of coverage could create pressure to settle. On the other hand, if the defendant lawyer cannot pay, any increased recovery based on a potential award of fees may be hollow, while requesting them may simply stir up a hornets nest by inviting a counterclaim for prevailing party attorney fees. These issues must be sorted out on a case-by-case basis.