We can do some rough estimates. Let's assume that bikes are made exclusively of metal, and that only the mass of the bike counts (no energy consumed in welding etc.). Looking up the carbon footprints of aluminium and steel we have:
So a bike weighing 10 kg would lead to somewhere between 8.6 (steel) and 200 (aluminium, refined using coal power) kg of CO2 emitted.
I couldn't quickly find reliable figures for carbon fibre or rubber, but metals are very likely to dominate based on the share of the mass in a bike, and the market share of bike frames.
Compared to an efficient car with 100 g/km of CO2 emissions, you'd have to replace 86 to 2000 km of driving with cycling for the emissions of making the bike to offset the emissions of using the car. This is a small enough distance that we can neglect other consumables. While not a direct comparison it's a helpful one—many people have a car and are being encouraged to get a bike and use it. Food makes things interesting. Strava claims I burn 26 Cal/km on a hilly ride. If I got that from eating beef alone (as a worst case), that would be something like 10 g of beef per km, or apparently 270 g CO2e per km. Of course we all know carbs are much better for powering our rides; typical sources are 10× better than beef.
Returning briefly to the original claim, when I've seen this claimed it tends to use low mileage figures, and neglect the fact that the metals in cars tend to be recycled (for Al this makes a big difference). It would make a good question at skeptics.se if you could source the claim more thoroughly.