I assume by 'phone' you mean mobile phone!
The short answer is yes. You can call yor existing mobile phone provider pay-as-you-go or contract and request a Port Authority Code, PAC code. This code is then given to the new network and you number transfer can begin.
Your existing provider is obliged to supply your PAC within 48 working hours from your request, this is specified in OFFCOM/OFFTEL/OTELO regulations on Mobile Number Portability, MNP.
In some cases you can even move to a new service provider on the same network, this relates to Vodafone and O2 contract customers only this is know as a migration.
Another thing to consider is what sort of coverage will youget on your new network. It is good to know that Virgin uses T-Mobile signal. BT Mobile use Vodafone signal now, Fresh (from carphone warehouse) uses T Mobile and 3, three, uses O2 signal when you cannot get a 3 signal.
Some networks require that the PAC code is given at point of connection, (Vodafone), others require it is given within a specific time, (O2 and BT Mobile both 14 days) and others will allow you to port at any time, (Orangte, T-Mobile, 3)
If you tell me what network you are on I can give greater detail.