The moves by North Lanarkshire Council come months after it revealed it could slash up to 1300 jobs in a bid to save £73million over three years.
The latest proposals have been announced by North Lanarkshire Leisure (NLL), the council's sports trust, which needs to cut £2m from its budget over three years.
Included in a consultation is a proposal to close the Iain Nicolson Recreation Centre, Moodiesburn, and reduce opening hours at the John Smith Pool, Airdrie; Kilsyth Pool and also the pool at Shotts Leisure Centre.
There is also a plan to close the pool at the Aquatec Leisure Centre, Motherwell.
A spokesman for NLL blamed the cuts on "financial pressures".
He added: "As a charitable sports trust, NLL is not immune to these efficiency pressures via significant reductions in the contract management fee from North Lanarkshire Council.
"The fee may be reduced by about £2m over the next three years, depending on the outcome of the £73m savings options consultation, and we are also facing increased cost pressures of £2m.
"This means we are being forced to make difficult decisions. One of these includes proposals to close the Iain Nicolson Centre.
"We are in discussions with staff and customers about this, which, if it goes ahead, would be implemented from March.
"However, given the excellent new facilities at Chryston High School and Cultural Centre we are confident most of the existing services will continue in the area."
But the plans were slammed as "retrograde" by the council's opposition SNP group.
Deputy group leader Tom Johnston, said: "This goes against all current thinking on healthy living and makes a joke of the idea of the Olympic legacy.
"We had the International Children's Games in Lanarkshire recently and this is not in the spirit of that event.
"Closing swimming pools is a retrograde step. We are totally against it."
Sean Hanlon, branch spokesman for Unison, which represents many of NLL's workers, also criticised the proposals.
He said: "Is the council's legacy, in the wake of the Olympic and the 2014 Commonwealth Games, to be known as a local authority that closed leisure sites?"
The consultation is also exploring the potential to remove senior leisure attendants at Wishaw Leisure Centre and the Tryst, Cumbernauld, and introduce changes to contracts for swimming teachers at all pools.
The consultation ends on December 14.