Yes, the safety has been approved by the FDA (U.S. Food and Drug Administration) and EFSA (European Food Safety Authority) as a food additive.

FDA
FDA considers the addition of MSG to foods to be “generally recognized as safe” (GRAS). It is the sodium salt of glutamic acid and FDA classified it in the list of “Special Dietary and Nutritional Additives” with amino acids and claimed it is a direct food additive and may be safely used as nutrients added to foods. (5)

EFSA
Monosodium glutamate (E 621) is authorised as a food additive in the European Union (EU) in accordance with Annex II to Regulation (EC) No 1333/2008 on food additives and categorized as “additives other than colours and sweeteners”  (6)

Re-evaluation In 2017
Safety

No adverse effects were observed in the studies of genotoxicity, carcinogenicity, neurotoxicity, reproductive and developmental toxicity.

ADI

In June, 2017, EFSA derived a group acceptable daily intake (ADI) of 30 mg/kg bw per day, expressed as glutamic acid.

Side Effects

EFSA also indicated the ADI that related with the MSG symptom complex (> 42.9 mg/kg), headache (85.8 mg/kg), blood pressure increase (150 mg/kg) and also insulin increase (> 143 mg/kg).(7)