The footage, filmed on 22 December, shows a dog chasing a fox over the wall. As the footage continues, the dog remains in pursuit of the fox.
Campaigners who reported the footage to police claimed a huntsman was above and behind them.
While traditional fox hunting is banned in Scotland, hunts in some forms are allowed.
The Buccleuch Hunt said it operated within the law at all times and described the allegations as "spurious" and "politically-motivated".
Campaigners have said it would be up to the courts to decide whether the hunt was illegal.
Robbie Marsland, from the League Against Cruel Sports, told BBC Scotland: "They say they're shooting foxes…we never see any guns being effectively used but what we do see is pictures like this of foxes being chased by hounds.
"We reported it to the police because it looks illegal to us but whether there's a successful prosecution will depend on the courts."
A statement from the Buccleuch Hunt said: "The Buccleuch Hunt operates within the law at all times to comply with the Protection of Wild Mammals (Scotland) Act. The hunt is absolutely clear that it was operating wholly within the law and in accordance with the Scottish Mounted Foxhound Packs Fox Control Protocol on the 22nd December.
"There is no evidence to support any claims to the contrary and the hunt urges animal rights activists to stop wasting police time by making spurious, politically-motivated allegations."
A Police Scotland spokesperson said: “We are aware of a report relating to fox hunting on land near Kelso on Wednesday, 22 December, 2021.
“Officers are awaiting further details from the complainer and enquiries will be carried out to establish the full circumstances.”
A Scottish government spokeswoman said: "Despite the ban on hunting introduced in 2002, we are aware there remains considerable public concern about fox hunting in Scotland and doubts about the operability of the legislation as it currently stands.
"It is our intention to put an end to this practice for good, and we have committed to bring forward legislation to do so before the end of the current parliamentary year. "