The La Liga giants are reportedly keen to avoid missing out on yet another highly rated centre-back.
The Catalans have been linked with a move for the highly rated defender previously, having scouted him extensively alongside Gerard Deulofeu, according to the Telegraph.
However, it would take big money to secure him from Goodison Park, with them already having rejected several offers from Chelsea in the summer.
They would not be put off by a big fee though, with them keen to avoid the mistake of missing out on a young, talented centre half again, according to Mundo Deportivo.
The club failed to secure deals for Brazilian duo Thiago Silva and Marquinhos in the past, with the duo now at Paris Saint Germain and rated among the best in the world.
Barcelona do not want that to happen again, and thus would pay big money to get Stones from Everton ahead of any potential rivals.
Everton would likely be hard to persuade into a deal regardless of the price on offer though, with Stones very much a part of their plans for the future.