The Everton midfielder enjoyed an impressive afternoon as he helped his side do the double over reigning Premier League Champions Leicester City this term.
Barkley, 23, was praised by the Blues boss afterwards, with Koeman glowing about the assist he provided for Romelu Lukaku's first goal of his brace.
But when asked about the midfielder's contract situation after the 4-2 win over the Foxes, Koeman repeated the uncompromising stance he first aired last month when asked about the situation.
He said: "We offer him a new contract and two possibilities: one is sign, if he doesn鈥檛 we need to sell the player.
"It is simple, it is not so difficult in my opinion."
Barkley only has a little over 12 months remaining on his current deal, with Everton keen to tie him down to extended terms.
The Blues are reluctant to let him enter the final year of his deal without a resolution, especially with the prospect of losing him for nothing in summer 2018.
Tottenham have been reportedly monitoring his progress this term.