In this paper, we are concerned with a class of chemotaxis-fluid models, which is a coupled system by parabolic-parabolic Keller–Segel equations and incompressible Navier–Stokes equations. We first establish the global existence of solutions with small initial data in critical Fourier–Besov spaces by a special type of iteration scheme. Then by introducing appropriate weighted functions based on carefully examining the algebraic structure of the system, we prove that under the conditions \(2\le p<\infty \) , \(2\le r\le q<\infty \) and \(\begin{aligned} \max \{\frac{1}{p}-\frac{1}{q},\frac{1}{p}-\frac{1}{r},\frac{1}{q}-\frac{1}{p},\frac{1}{r}-\frac{1}{q}\}<\frac{1}{3}, \end{aligned}\) there exist two positive constants \(K_{0}\) and \(k_{0}\) such that if the initial data \((u_{0},n_{0}, c_{0})\) satisfies \(\begin{aligned} \Vert u_{0}\Vert _{{F{\dot{B}}}^{2-\frac{3}{p}}_{p,1}} +K_{0}\Vert n_{0}\Vert _{{F{\dot{B}}}^{1-\frac{3}{q}}_{q,1}}\exp \big \{K_{0}\Vert c_{0}\Vert _{{F{\dot{B}}}^{3-\frac{3}{r}}_{r,1}}\big \}\le k_{0}, \end{aligned}\) then the coupled chemotaxis–fluid equations admits a unique global solution.
