The upcoming launch of the James Webb Space Telescope (JWST) will dramatically increase our understanding of exoplanets, particularly through direct imaging. Microlensing and radial velocity surveys indicate that some M dwarfs host long-period giant planets. Some of these planets will likely be just a few parsecs away and a few astronomical units from their host stars, a parameter space that cannot be probed by existing high-contrast imagers. We studied whether the coronagraphs on the Mid-infrared Instrument on JWST can detect Jovian-type planets around nearby M dwarfs. For a sample of 27 very nearby M dwarfs, we simulated a sample of Saturn-Jupiter-mass planets with three atmospheric configurations and three orbital separations, observed in three different filters. We found that the f1550c 15.5 {mu}m filter is best suited for detecting Jupiter-like planets. Jupiter-like planets with patchy cloud cover, 2 au from their star, are detectable at 15.5 {mu}m around 14 stars in our sample, while Jupiters with clearer atmospheres are detectable around all stars in the sample. Saturns were most detectable at 10.65 and 11.4 {mu}m (f1065c and f1140c filters), but only with cloud-free atmospheres and within 3 pc (six stars). Surveying all 27 stars would take <170 hr of JWST integration time, or just a few hours for a shorter survey of the most favorable targets. There is one potentially detectable known planet in our sample: GJ 832 b. Observations aimed at detecting this planet should occur in 2024-2026, when the planet is maximally separated from the star.