MIDDLEBORO — Two select board members expressed disappointment with the school budget at Monday night’s meeting, with one ultimately deciding not to support it.

Select Board Chair Mark Germain and Select Board member Brian Giovanoni initially refused to support the school’s $40 million budget, saying that cuts required to reach that amount left the district in an unsustainable financial position for next year.

Superintendent of Schools Carolyn Lyons had originally proposed a budget of about $46 million. That amount was cut to $40.5 million to reach a figure that Town Manager James McGrail said the town could work with.

Giovanoni and Germain questioned the cuts made to reach that $40.5 million, noting they came from sources that were unsustainable going forward, including revolving funds and grants.

Taking away from these funding sources could leave the schools on a “fiscal cliff,’’ Giovanoni said, with fewer resources to rely on for next year.

Noting that McGrail could accept the budget, select board members Neil Rosenthal and Thomas White gave their support although Rosenthal expressed reservations.

“They’re digging the hole,’’ he said. “They have to dig themselves out.’’

With the vote at a 2-2 tie, the motion failed. Germain then agreed to support the entire budget to move the process along. Giovanoni did not change his mind and said that would “not be surprised if someone makes a motion (at Town Meeting) to cut $1 million from the budget.’’

The ultimate decision on whether to approve the budget will be made by Town Meeting voters April 22.

McGrail said discussions need to begin as early as this summer to start considering next year’s school budget and its potential challenges. If there are going to be problems, he said, “I’d rather know in July than in December.’’