The Goodison Park club cashed in on top scorer Romelu Lukaku when they sold him to Manchester United in the summer, taking Wayne Rooney in return, and they also signed Sandro Ramirez from Malaga.
But Barton feels manager Ronald Koeman was expecting further reinforcements, and has revealed he understands Kenwright stopped any more spending.
Co-hosting the Alan Brazil Sports Breakfast alongside Jeff Stelling, Everton fan Barton said: “I think Koeman was trying to get in another striker, and if it had been down to him he would have. They wanted and identified targets to try to bring in.
“I think the issue was Kenwright, from what I gather. He’s still the custodian in terms of being chairman. Koeman and [Farhad] Moshiri thought they would have money to spend but Kenwright was wanting to watch the budget, and didn’t want to over-spend.”