I think this is not correct. (Although I am a practising solicitor, I do not handle my firm's accounts, which is why I would not be an authority on this). As best I recall, a solicitor must account to his client for interest once it goes beyond a certain figure (maybe €100 - I can't remember).
However, the interest is at the demand deposit rates appropriate to the individual sum of money (as the money is and must be available on demand) so if the solicitor has negotiated a better rate with the bank (as he\she has lots of such sums on deposit) the surplus belongs to the solicitor.