The Scotland centre-back is in the final months of his deal at Tynecastle Park.
Hearts have put an offer to the player but were aware they would face stiff competition.
Not only have Celtic and Rangers been linked with a move but also a number of Championship clubs in England, including Stoke City where John's brother Harry is a key player and Nottingham Forest.
Blackburn are the first to make a noteworthy move with Rovers managed by former Celtic and Hibs boss Tony Mowbray who will be aware of the potential in the Scottish market.
They are also strong admirers of Aberdeen forward Ryan Hedges who is also out of contract at the end of the season and is expected to move south and could do so in January.
For Souttar, even if he signs a pre-contract agreement it would take a significant offer for Hearts to sell in the transfer window.
The 25-year-old is a key player, making 22 appearances so far with his form having earned him a return to the international fold.
“We are still trying to get John signed up,” Neilson said.
"There is no doubt he is a top player and we just hope that even if he doesn't [sign], he is here for the rest of the season.”