The 29-year-old, a boyhood Toon fan, is in the last 18 months of his contract.
Brighton are not actively seeking to sell Burn this transfer window and an opening offer of £8million from the Magpies was swiftly rejected.
However, with their pursuit of Sevilla's Diego Carlos dead in the water, Newcastle could yet return with a higher offer.
Time is running out for the Magpies to land a centre-back this month, which means clubs may exploit their newly-found wealth and desperation to stay in the Premier League.
But the Seagulls may also be tempted to seize an opportunity to cash in on Burn, rather than let the Englishman run down his contract.
The defender has been a key player for Graham Potter's side this season, striking up an excellent partnership with Adam Webster in recent weeks.
Burn is comfortable in the middle of a back three, or in a centre-back pairing, making him a versatile player with Premier League pedigree - exactly what Newcastle need.
Some reports suggest they are also interested in Brighton's Webster, but so far their raid of the Amex is following a familiar pattern to their other transfer bids.
The Magpies have struggled to add to their early additions of Kieran Trippier and Chris Wood - and earlier on Wednesday, Lyon denied they had agreed a fee for midfielder Bruno Guimaraes.
Speaking from the club's training base in Saudi Arabia, manager Eddie Howe told NUFC TV: "I think they have no news to tell anybody at the moment, but I'm keeping regularly updated and finger on the pulse as to what is happening.
"As we know, it is a difficult window and we just hope we can get a couple more additions to boost the squad."