Hi,
The question is not just what diameter bore but also what depth? Some details on that will make a suggestion much easier.
For up to 12x diameter long bores of Ø4.9mm and up I've been using some kennametal products. The inserts are proprietary so not cheap but they have sizes from a shank of 4mm,5mm, 6mm that all use the same tips. I've got the solid carbide shank bars second hand and got the tips from fg metcalfe and son. All the major vendors have their own range for this type of thing.
For below Ø4.9mm I've been using some of the Carmex "tiny tools". Same as Old School's reference supplies.
Steve