The 30-year-old, who made 37 appearances for West Ham in the Premier League last season, returned north for a medical with the Championship side after being left out of the Hammers’ Europa League squad to face Birkirkara.
Downing, capped 35 times for England, began his career with his home-town club before earning a £12m move to Aston Villa in July 2009. He then swapped life in the Midlands for Liverpool, who paid £20m for his services two years later.
The midfielder enjoyed a resurgence in form after heading to West Ham for £5m in August 2013, but appears not to be part of new manager Slaven Bilic’s plans.
He is one of two likely arrivals at Boro after they also agreed a fee to sign Uruguay international striker Cristhian Stuani from Espanyol.
The 28-year-old, who scored 16 goals in 48 appearances in all competitions last season, is set to join for a reported £2.8m, with half of the fee going to his former club Reggina.
A statement from Espanyol read: “The Uruguay striker will join Middlesbrough once he passes a medical and signs his contract with the English club.”