Over the last five years, the Cleveland Museum of Art has been at work on one of the largest building programs of any art institution in the country, a $350 million project that has been unveiled in sleek new stages and will be completed by 2013, adding 35,000 more square feet of gallery space.

But the museum has also been building in less visible ways and is set to announce on Monday the acquisition of two high-profile ancient artifacts that seem certain to draw attention not only to the institution’s expansion but also to the complicated long-running debate about antiquities collecting by museums.