The Athletic says there is now a growing belief that Jackson will leave Chelsea before the summer market shuts.
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The Senegal international has been linked with Manchester United and Newcastle.
Jackson has kept the door open to leaving the Blues this summer when discussing his future publicly.
Asked about his future, Jackson said last month: "I’m a Chelsea player. The future belongs to God. If there are any changes, you’ll know. For now, I’m focused on my preparation.’
"It’s football, there are ups and downs. Last season we won two trophies. There were difficult moments, moments of glory. Every season is like that. But the most important thing is to stay focused, to learn, and to grow in the face of adversity."