In this article, we explore the mapping and boundedness properties of linear and bilinear fractional integral operators acting on Lebesgue spaces with large indices. The prototype ν -order fractional integral operator is the Riesz potential md5=8e07cf170075e093857e2fac110ee52e" title="Click to view the MathML source">Iν, and the standard estimates for md5=8e07cf170075e093857e2fac110ee52e" title="Click to view the MathML source">Iν are from md5=75faae3c94ea2f4c169ab8b9d78be339" title="Click to view the MathML source">Lp into md5=f3cdd959a191d1f5b61df2383d595453" title="Click to view the MathML source">Lq when md5=e9e6c000b85fd9bd767a409f221053a0"> and md5=5965d40a472d112f022525becde64ccb">. We show that a ν -order linear fractional integral operator can be continuously extended to a bounded operator from md5=75faae3c94ea2f4c169ab8b9d78be339" title="Click to view the MathML source">Lp into the Sobolev-BMO space md5=b95ab6851a303e49a0aa16b7b09c57ed" title="Click to view the MathML source">Is(BMO) when md5=f8d572a69a02d03fe733edcb49a579b0"> and md5=f312c8b3339a849ec7e3d09273afcf06" title="Click to view the MathML source">0≤s<ν satisfy md5=fc8182b771c8cf17c2eaa6320d3645a2">. Likewise, we prove estimates for ν -order bilinear fractional integral operators from md5=bf874ff9081478b59b6451ec436a2ab6" title="Click to view the MathML source">Lp1×Lp2 into md5=b95ab6851a303e49a0aa16b7b09c57ed" title="Click to view the MathML source">Is(BMO) for various ranges of the indices md5=002fae8aa964dabde87c643f0155f61d" title="Click to view the MathML source">p1, md5=c530fad0e79aa63a6786fd8d9bad09b6" title="Click to view the MathML source">p2, and s satisfying md5=4693c118743e88503574a05eb50fa113">.