FIG. 1 is a front perspective view of a bicycle basket supporting bracket showing my design; FIG. 2 is a rear perspective view thereof; FIG. 3 is a rear elevational view thereof; FIG. 4 is a front elevational view thereof. FIG. 5 is a right elevational view thereof; FIG. 6 is a left elevational view thereof; FIG. 7 is a top plan view thereof; and, FIG. 8 is a bottom plan view thereof.