The 1.67-acre Arastradero Road property in question is owned and occupied by the independent Bowman International School, which reportedly is in the market for a larger site.
Any deal with Palo Alto would depend on Bowman securing an alternate location.
"We're in preliminary discussions with Bowman School leadership, and there are issues certainly around price," Superintendent Kevin Skelly said.
"Land in Palo Alto is not inexpensive, but when you consider the ability to acquire a piece of property that's next to our smallest middle school, that has a value to the district. It gives us the ability to expand our middle school capacity in ways that building a fourth middle school would be considerably more expensive and difficult to find."
Current enrollment in Palo Alto's three middle schools, at 2,733, is fast approaching full capacity — without portable classrooms — of 2,950. Portable classrooms could expand that by several hundred students, officials said.
A Bowman acquisition would allow Terman to grow from its current capacity of 750 to a capacity of 1,100, matching the sizes of Jordan and JLS middle schools.
Traffic disruptions would be minimal since Bowman already is home to 225 students, nearly all of whom are driven to school, Skelly said.
Seismic and other retrofitting would be necessary to bring the 12-year-old Bowman infrastructure to the strict standards required of all California public schools by the Division of State Architect, school officials said.
School planners have said the new space will be needed in eight years — when middle-school enrollment is projected at 3,148 — but several school board members have advocated for a quicker timetable.
"I believe we need to have something in place five to eight years out, not eight years," member Melissa Baten Caswell said Tuesday.
If current trends continue, a fourth middle school eventually would be needed.
Board members instructed Skelly to pursue talks with Bowman but also to continue a broader search for space.
"In the 1980s the district lost something around 50 acres of school property (when schools were closed and sold for housing), so any time we identify land adjacent to an existing site it provides permanent flexibility ... to the district and families and residents," board member Barb Mitchell said.
"There's no question that Palo Alto is going to continue to grow, and there's going to be an interest in having public services to grow somehow in relation to housing, so I support this."