Mr Dalton said as the council had already set aside "substantial funding" toward the railway in its Draft Annual Plan, "they were the obvious choice" to secure its future.
Mr Yule said it was also agreed a business case for the railway should be made by June 30, next year.
"We've been debating the railway's future for five or six years and I think June 30, is a good time for finality to be given around that."
Council chairman Fenton Wilson said while the end use of the railway was still being considered, a decision to lease it was "now urgent".
"We all support the process to secure access to the rail corridor, whatever its use, as soon as possible."