Interest rates can vary wildly for different current accounts but as a general rule, they will always be lower than the rates you could expect from a savings account.
However, since the introduction of legislation making it easier to switch between current accounts has forced the market to become more and more competitive, we have been seeing more and more current accounts with interest rates going as high as 5% AER (annual equivalent rate).
There is often a catch though. Some of the highest interest current accounts will only pay out interest for one year, for example. Many current accounts will also have an upper limit placed on the balance on which they will pay interest, or will scale down the interest rate as the balance increases. So for example, your account might pay 2% interest on anything up to £1000, and then 1.2% on anything from £1,001 to £1,500, and then no interest on anything over £1,500.