This paper deals with the asymptotic behavior of solutions to the initial-boundary value problem of the following fractional p-Kirchhoff equation: ut+M([u]s,pp)(−Δ)psu+f(x,u)=g(x)in Ω×(0,∞),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ u_{t}+M\\bigl([u]_{s,p}^{p}\\bigr) (-\\Delta )_{p}^{s}u+f(x,u)=g(x)\\quad \\text{in } \\Omega \\times (0, \\infty ), $$\\end{document} where Omega subset mathbb{R}^{N} is a bounded domain with Lipschitz boundary, N>ps, 0< s<1<p, M:[0,infty )rightarrow [0,infty ) is a nondecreasing continuous function, [u]_{s,p} is the Gagliardo seminorm of u, f:Omega times mathbb{R}rightarrow mathbb{R} and gin L^{2}(Omega ). With general assumptions on f and g, we prove the existence of global attractors in proper spaces. Then, we show that the fractal dimensional of global attractors is infinite provided some conditions are satisfied.