Inflammatory bowel diseases (IBD) and acute graft-versus-host disease (GVHD) are associated with persistent intestinal dysfunction preceded by gut bacterial dysbiosis. There are limited data on intestinal bacteriophages in these conditions. The aim of the present work was to detect associations between dominant intestinal bacteria by means of 16S rRNA gene sequencing, and some clinically significant viruses detected with a customized primer panel for NGS-based study. The clinical group included patients with Crohn’s disease (IBD, n = 9), or GVHD (n = 6) subjected to fecal microbiota transplantation (FMT) from healthy donors. The stool specimens were taken initially, and 5 times post-FMT until day 120. Using NGS approach, we have found a higher abundance of Proteobacterota phylum in GVHD, especially, at later terms post-FMT. Moreover, we found an early increase of Klebsiella and E. coli/Shigella abundance in GVHD, along with decreased relative content of Faecalibacterium. Upon evaluation of intestinal phageome, the relative amount of Caudoviricetes class was higher in GVHD. A significant correlation was found between Proteobacteria and Caudoviricetes, thus suggesting their association during the post-FMT period. Moreover, the relative amounts of five Caudoviricetes phage species showed distinct correlations with Klebsiella and Enterococcus ratios at different terms of FMT. In conclusion, parallel use of 16S rRNA gene sequencing and targeted NGS viral panel is a feasible and useful option for tracing specific viral strains in fecal microbiota. The developed array of viral primers may be extended to detect other phages infecting the clinically relevant bacteria.