The possibility that primordial black holes (PBHs) form a part of dark matter has been considered for a long time but poorly constrained in the $1-100~M_{odot}$ (or stellar mass range). However, a renewed special interest of PBHs in this mass window was triggered by the discovery at LIGO of the merger events of black-hole binaries. Fast radio bursts (FRBs) are bright radio transients with millisecond duration and high all-sky occurrence rate. Lensing effect of these bursts has been proposed as one of the cleanest probes for constraining the presence of PBHs in the stellar mass window. In this paper, we first investigate constraints on the abundance of PBHs from the latest FRB observations for both the monochromatic mass distribution and three other popular extended mass distributions (EMDs). We find that constraints from currently public FRB observations are relatively weaker than those from existing gravitational wave detections. Furthermore, we forecast constraining power of future FRB observations on the abundance of PBHs with different mass distributions of PBHs and different redshift distributions of FRBs taken into account. Finally, We find that constraints of parameter space on EMDs from $sim10^5$ FRBs with $overline{Delta t}leq1 ~rm ms$ would be comparable with what can be constrained from gravitational wave events. It is foreseen that upcoming complementary multi-messenger observations will yield considerable constraints on the possibilities of PBHs in this intriguing mass window.
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