The simple answer to this is no.
Generally speaking, only a Court or tribunal can order one party to pay the other party’s legal costs. Before a claim is issued, the Court has no jurisdiction to do this. Once a claim is issued, then the pre-action cost incurred can be claimed. However, someone would be ill advised to issue their claim without going through the expected attempts to reach an agreement first.
Special types of offer, called Part 36 Offers, can sometimes be made and, if accepted, would entitle a party to claim their legal costs up to the date of acceptance.
There may also be a contractual right to claim the costs of dealing with the dispute, depending on the terms of any contract involved.