It is understood a Russian taskforce, including the aircraft carrier Admiral Kuznetsov, is on a course to sail through the North Sea and then the English Channel.
Two Russian corvettes, which are travelling north towards the UK from the direction of Portugal, are also set to be watched by the Navy.
The northern fleet passed the Norwegian city of Bergen on Wednesday and will sail through the English Channel, past Gibraltar and into the Mediterranean Sea towards the Syrian coast, according to a senior Nato diplomat quoted by Reuters.
The anonymous source said the Russian warships are carrying fighter bombers likely to bolster the attack on the besieged Syrian city.
The diplomat said: “They are deploying all of the northern fleet and much of the Baltic fleet in the largest surface deployment since the end of the Cold War.
“This is not a friendly port call. In two weeks, we will see a crescendo of air attacks on Aleppo as part of Russia’s strategy to declare victory there.
“With this assault, it should be enough to allow a Russian exit strategy if Moscow believes Assad is now stable enough to survive.”
Type 45 destroyer HMS Duncan sailed from Portsmouth on Tuesday to monitor the Kuznetsov task group, which is heading south from the Norwegian Sea towards the North Sea.
A Ministry of Defence spokesman said: “When these ships near our waters we will man-mark them every step of the way. We will be watching as part of our steadfast commitment to keep Britain safe.”
Type 23 frigate HMS Richmond has also escorted the group in the Norwegian Sea, north of the UK between Iceland and Norway.
Another Type 45 destroyer, HMS Dragon, is also due to sail to meet the two Russian corvettes travelling north towards the UK from the direction of Portugal.
It is understood that shadowing ships is regular business for the Royal Navy, being supported by RAF surveillance aircraft on this occasion.
Navy ships will continue to escort the Russian task group and provide a visible presence south through the North Sea and English Channel as the Russian fleet sails through, the MoD said.
The UK is working alongside Nato partners to monitor the warships.