The England international was given permission to hold talks with City by the Midlands club earlier in the day and his transfer appears to have now taken a major step closer to completion.
Barry was the subject of intense interest from Liverpool last summer, but the deal fell through with the Anfield outfit unwilling to meet Villa's £18million asking price.
City are reported to have agreed a fee of £11million with their Premier League rivals, with Barry's current contract due to expire at the end of next season.
The midfielder had previously stated he was keen to sample UEFA Champions League football, with Villa's top-four challenge this season fading as Martin O'Neill's men eventually finished sixth to secure a Europa League berth.
City will not be playing in Europe next term but Mark Hughes' side have ambitious plans for the future given the financial backing of their owners and have set their sights on challenging the Premier League's leading clubs.