The 20-year-old shot into the limelight last month when he was handed a shock senior debut against Tottenham in the Premier League.
And Travers rose to the task, keeping a clean sheet and being named man-of-the-match in the 1-0 victory.
That saw him keep his place in the side for the final-day defeat at Crystal Palace and also earned him a spot in the Republic of Ireland squad for their Euro 2020 qualifiers against Denmark and Gibraltar.
However, the young stopper was forced to withdraw after fracturing his thumb in training, meaning he has been forced to wait to make his international bow.
But the report claims that Travers' performances for Cherries have alerted newly-promoted Norwich and Championship side Leeds.
Travers signed a long-term contract with Cherries last summer but the club are said to be considering offering him a new and improved deal.
The goalkeeper is keen to get some more first-team football under his belt as he looks to return to Mick McCarthy's Ireland squad.
