The Yorkshire Post understands that the 27-year-old striker has been in talks with the Bedfordshire outfit today.
The former Fulham forward could be confirmed as a Hatters player tomorrow, pending the successful completion of his medical. The Hemel Hempstead-born player started his career at Kenilworth Road as a youngster.
Woodrow, one of the highest earners at Oakwell, has been linked with several clubs already this summer, including Ipswich Town, Portsmouth and Sheffield Wednesday.
Woodrow endured an injury-hit second half of last season - after undergoing knee surgery in December. But prior to that, the forward was a consistent scorer for the Reds for the previous three seasons.
Barnsley are needing to cut costs in the aftermath of relegation with the club faced with a shortfall of £7m to £8m following relegation.
Speaking earlier this month, Barnsley chief executive officer Khaled El-Ahmad hinted at departures from the club of some major players following the club's relegation to League One.
El-Ahmad continued: “Without speaking about specific names, I think it is a natural progression.
“You get relegated, some have been here a long time, some see their contracts expire soon.
“It is a mutual discussion both ways, but ultimately they are under contract and the club controls their destiny.”
Barnsley, who started pre-season training on Monday under new head coach Michael Duff, have brought in two players already in the shape of ex-Celtic and Bolton midfielder Luca Connell and New Zealand international keeper Jamie Searle.