The Daily Telegraph claims Tottenham are likely to submit an opening bid of £20m ($26m), plus £8m in add-ons, for Barkley and remain favourites to sign him, but that Chelsea are preparing a bid of their own.
Barkley has entered the final year of his contract at Goodison Park and is unlikely to sign an extension, with Everton manager Ronald Koeman confirming last month that the midfielder wanted to leave for "a new challenge".
The Toffees valued the England international at around £50m at the start of the summer, but their negotiating position has been weakened by the player suffering a hamstring injury.
The Merseyside club have already brought in Gylfi Sigurdsson from Swansea City in a £45m deal as a replacement for Barkley.
Barkley, 23, scored five goals in 36 league appearances for Everton last season. He has been ruled out for three months after tearing a hamstring in training earlier in August.
"We made a really good offer to him to sign a new contract, he declined that contract and told me that he is looking for a new challenge," Koeman was reported as saying by the Guardian in July.
"OK, he is not in Everton's future any more. It is his decision. I need to respect that and see what happens. What I heard from the board at the moment is that there is no offer on the table for Ross.
"Everybody knows what the situation is. He told me at the end of last season. He has one more year on his contract and we work with respectful people."