How much air, by mass, enters an average CFM56 turbofan engine cruising per minute?
The CFM56-5C, the largest CFM56, powers A340. Its thrust is about 145 kN at takeoff. It has an air mass flow of 485 kg/s (source).


CFM56-5C, source.

The Trent XWB (A350 XWB), a larger engine developing 430 kN at takeoff, has an air mass flow of 1,440 kg/s (source).

T/O vs cruise power and fuel consumption

According to this document, the CFM56-5C T/O thrust is 150 kN and the cruise thrust is 40 kN (other sources states 30 kN), 27% of the T/O thrust. Related fuel consumption are 2.3 and 0.5 kg/s, meaning cruise fuel consumption is 22% of the T/O fuel consumption.

Air mass vs fuel mass

For a fuel consumption of 22% of the T/O consumption, it can be assumed relatively safely that the air flow is in proportion, so we can roughly estimate air mass flow in cruise to be about 485 * 0.22 = 107 kg/s. This is supported by figures found online in non authoritative sources, like page 245 of this book: