North Kesteven District Council's executive committee said Sleaford's swimming pool could no longer meet "customer standards and expectations".
Councillors said the investment would enhance the centre and ensure its future. The full council will make a final decision next month.
In 2010, the authority tried but failed to secure outside funding from private investors.
The money for the refurbishment will mostly come from the council with some contribution from the Amateur Swimming Association.
The building will be remodelled with improvements to the changing rooms and gym area.
It is expected to close late next year for the work to be carried out with a reopening expected in December 2013.
Council leaders have voted to invest £2.5m in a Lincolnshire swimming pool.