If not, we’d suggest finding out now. If it’s not, it’s time to move your money. And right now many of the best interest rates are for Cash ISAs which do allow transfers in from existing ISAs. Here we're going to look at the best Cash ISAs that allow transfers in.We've included accounts that are tied to popular current accounts and those that have tiered interest rates.f you’d prefer to tie up your savings for a year or more, you could usually earn a far higher rate of interest with a fixed rate Cash ISA. Here are the top-paying fixed-rate Cash ISAs across a range of timespans that accept transfers. Account. Interest rate (AER).