It's understood Boro are putting the finishing touches to the move tonight.
Joseph was expected to join Boro at the start of last week, but the move stalled due to a complication.
However, despite reports at the end of last week claiming the move was off, it's understood that was never the case and the two clubs remained in discussions. And a new agreement has now been reached.
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The signing is expected to be finalised in the next 24 hours, which might mean Joseph links up with his new teammates in Portugal before the week is out.
The arrival of Joseph will be a big boost for Boro and will further strengthen the forward line after Myles Peart-Harris last week became the first signing of the summer.
Joseph is a versatile forward player, but it's understood Boro believe he's best equipped to play as a No.9.
Despite only starting 27 of Hull's 46 Championship games last season, the 24-year-old scored eight goals and provided five assists.
That tally of goal contributions was only bettered by two Boro players last term, Morgan Whittaker and Tommy Conway.
Hull were reluctant to sell Joseph this summer but initially agreed to a deal with Boro because they were in a race against time to raise £6m to ensure they didn't fall foul of financial regulations, which could have resulted in a six-point deduction in the Premier League.
They raised the required funds despite Joseph's move to Boro initially falling through, but the move to the Riverside is understood to have been hugely appealing to the forward. Personal terms were already agreed last week.
Boro will look to further strengthen their forward line with at least one more No.9 and other versatile forwards in the weeks to come.
They want to sign at least two midfielders this summer, a left-back and a senior goalkeeper.
Boro are stepping up their preparations for the coming season in Portugal this week before their first public-facing friendly away at Celtic a week on Saturday.