The internal structure of the superconducting state in Sr2RuO4 remains elusive at present, and exhibits evidence for time-reversal symmetry breaking. Recent muon spin relaxation measurements under uniaxial strain have revealed an increasing splitting between the superconducting critical temperature T-c and the onset of time-reversal symmetry breaking TTRSB with applied strain (Grinenko et al., arXiv:2001.08152). In addition, static magnetic order is induced by the uniaxial strain beyond similar to 1 GPa, indicating that unstained Sr2RuO4 is close to a magnetic quantum critical point. Here we perform a theoretical study of the magnetic susceptibility and the associated pairing structure as a function of uniaxial strain. It is found that the recent muon relaxation data can be qualitatively explained from the perspective of spin-fluctuation mediated pairing and the associated strain dependence of accidentally degenerate pair states in unstrained Sr2RuO4. In addition, while unstrained Sr2RuO4 features mainly (27 pi/3, 27 pi /3) magnetic fluctuations, uniaxial strain promotes (pi, +/-pi/2) magnetic order.