The private meeting of the five rural TDs, held on Tuesday night, was also attended by the two Healy-Rae brothers.
It is understood that Michael Healy-Rae asked the other six non-aligned TDs at the meeting who would vote for Mr Kenny as taoiseach.
A show of hands resulted in support being pledged by newly elected Clare TD Michael Harty, Roscommon-Galway TD Denis Naughten, as well as Michael Healy Rae. It is also thought that Galway West TD Noel Grealish could be persuaded to support Mr Kenny for the next vote.
Sources said that, if the result of the private meeting is reproduced at the next vote for Mr Kenny, it would give him at least three to four extra votes in his bid to form a minority government.
It is thought the other two TDs in the rural alliance, Tipperary’s Mattie McGrath and Cork South West’s Michael Collins, are unlikely to vote for Mr Kenny.
Separately, Danny Healy-Rae yesterday said Fine Gael and Fianna Fáil are both just interested in the next general election. It is understood that he is unlikely to support Mr Kenny or a minority government.
The support of three to four rural TDs for Mr Kenny’s efforts to form a minority government would greatly aid his position if the six-person Independent Alliance gets behind him.
The alliance is waiting until the government talks are finished between the two big parties before making a final decision. One member, Waterford TD John Halligan, is reluctant to commit to supporting a Fine Gael minority government without funds for a full-time cardiac care unit for his local hospital.
Alliance sources say they are confident of their local demands being addressed and that a deal hangs upon what Fianna Fáil demands.
Support from the alliance, as well as the four rural TDs, would give Mr Kenny up to 62 backers at the next vote in the Dáil, a position which would make him taoiseach, if Fianna Fáil abstains in any vote.