The Tykes were relegated from the Championship last season while the Hatters reached the play-offs.
Morris scored nine goals and provided three assists in 30 appearances in all competitions for Barnsley last term.
In an update on Monday night, The Yorkshire Post's Leon Wobschall said that the player's move to Luton was nearing completion, after an offer had been rejected from another Championship club.
Last month, Luton signed 27-year-old striker Cauley Woodrow for an undisclosed fee with Morris set to reunite with his former Oakwell teammate.
Morris started his career with Norwich City but spent the majority of his time with the Canaries on loan at other clubs, appearing for Hamilton Academical, Shrewsbury Town and MK Dons alongside his spells in Yorkshire.