The Selenga is one of the crucial transboundary rivers of the semi-arid Northern Eurasia belt. The Selenga basin is located in Mongolia and Russia, and it is 83.4% of the Lake Baikal catchment area. Atmospheric precipitation is the primary source of the river supply, most of its amount (of about 450 mm per year, about 70% of the annual) falls like rain from June to August (about 70% of the annual). In the present paper, the relationship between the heaviest rains (HR) around Selenga River basin in July (above 90th percentile) and Rossby wave breaking (both cyclonic and anticyclonic type, AWB, and CWB) was examined. The total number of HR events from 1982 to 2019 was 83. For each event, the synoptic analysis and automatic detection of breaking based on potential vorticity from 2 to 9 PVU on the 350 K were utilized. In most cases (86%) of HR, events were accompanied to the RWB. It was revealed that waves are propagating along the subtropical jet (PVU overturning on 350 K) were the most important. Precipitation was observed both for the period of amplitude growth and period of waves breaking (CWB or AWB) up to the barotropic stage. However, CWBs on the subtropical jet stream that occurred east to Lake Baikal were observed in most of the HR events. The high vertical instability and precipitable water were characteristic of these CWBs. The case study is shown that the high precipitable water in part was due to the “East Asian summer monsoon” northward jump.