Yes, but not at the same time — magnesium can bind levothyroxine in the gut and reduce how much you absorb, so separate them by at least 4 hours and confirm timing with your pharmacist.
The risk here is reduced effectiveness, not a dangerous reaction: magnesium is a polyvalent (multi-charged) mineral that can bind levothyroxine in the stomach and form an insoluble complex that the gut absorbs poorly, which can leave your thyroid undertreated. The FDA-approved Synthroid (levothyroxine) label lists antacids containing aluminum and magnesium hydroxides among the agents that decrease levothyroxine absorption and may result in hypothyroidism, and directs that levothyroxine be given at least 4 hours apart from them. The NHS says antacids, calcium salts and iron salts can reduce the amount of levothyroxine your body takes in when taken at the same time, and MedlinePlus gives the same at-least-4-hours spacing advice for similar binding products like calcium and iron. The clearest documented case is magnesium-containing antacids, and the same 4-hour spacing is a sensible precaution for any oral magnesium supplement.
Take levothyroxine once daily on an empty stomach, 30 to 60 minutes before breakfast, with water; move any magnesium supplement or magnesium-containing antacid to at least 4 hours later (the same rule applies to calcium and iron). Keep your timing consistent day to day. Watch for signs your thyroid dose is no longer working if you start taking magnesium close to your pill — tiredness, weight gain, feeling cold, constipation, or low mood — and ask your doctor to check your TSH; the label recommends TSH monitoring when absorption may be affected. Tell your doctor or pharmacist about every supplement and antacid you take, and ask the pharmacist to confirm the exact gap for your specific magnesium product before changing your routine. Do not stop or change your levothyroxine dose on your own.
This is general reference, not medical advice, and not a guarantee of safety. Interactions depend on your doses, health conditions, and other medicines. Always confirm with your pharmacist or doctor before combining products, and follow the dosing on each label.