Recent observational studies of intermediate-age star clusters (SCs) in the Large Magellanic Cloud (LMC) have reported that a significant number of these objects show double main-sequence turn-offs (DMSTOs) in their colour-magnitude diagrams (CMDs). One plausible explanation for the origin of these DMSTOs is that the SCs are composed of two different stellar populations with age differences of ∼300 Myr. Based on analytical methods and numerical simulations, we explore a new scenario in which SCs interact and merge with star-forming giant molecular clouds (GMCs) to form new composite SCs with two distinct component populations. In this new scenario, the possible age differences between the two different stellar populations responsible for the DMSTOs are due largely to secondary star formation within GMCs interacting and merging with already-existing SCs in the LMC disc. The total gas masses being converted into new stars (i.e. the second generation of stars) during GMC-SC interaction and merging can be comparable to or larger than the masses of the original SCs (i.e. the first generation of stars) in this scenario. Our simulations show that the spatial distributions of new stars in composite SCs formed from GMC-SC merging are more compact than those of stars initially in the SCs. We discuss both advantages and disadvantages of the new scenario in explaining fundamental properties of SCs with DMSTOs in the LMC and in the Small Magellanic Cloud (SMC). We also discuss the merits of various alternative scenarios for the origin of the DMSTOs.