David Cameron told the G7 summit in Japan that the UK was ready to take an "active leadership role" in helping Libya deal with trafficking.
The UK currently has one survey vessel, HMS Enterprise, operating in the area.
For the UK to send another warship, the EU will need to extend the mandate of its current naval operation.
UK officials at this week's meeting of G7 industrialised nations said they would seek UN approval so that the new warship could also seize boats suspected of smuggling arms to fighters of so-called Islamic State in Libya.
Last Sunday the new Government of National Accord (GNA) in Tripoli asked the European Union for more help in training its navy and coastguard to stop the trafficking of migrants across the Mediterranean.
It is thought this may soon be followed by a request for international ships to operate in Libyan waters.
If this request is received, the UK will seek the extension of the EU's Operation Sophia mission to tackle people-trafficking in the central Mediterranean, as well as a Security Council resolution at the UN enabling its forces to assist in the interception of arms shipments.
This would move the UK another step closer to direct military involvement in the Libyan conflict, BBC diplomatic correspondent James Landale says.
This week the UK sent four military planners to the Rome headquarters of Operation Sophia, with the goal of preparing a plan to improve the effectiveness of the Libyan coastguard.
Downing Street did not say what type of Royal Navy ship would be sent and declined to discuss what evidence it had for illicit arms movements from the Libyan coast.
Mr Cameron first put forward the idea of extending Operation Sophia into Libyan territorial waters at a Brussels summit of the EU in March.
He warned that the number of migrants attempting to cross via the central Mediterranean could be expected to swell once the alternative route through Turkey, Greece and the western Balkans had been closed.
The UK is set to send a second Royal Navy ship to the Mediterranean to help tackle people trafficking and arms smuggling in Libya, officials say.