Catalysis of the reaction: L-methionine R-oxide + thioredoxin = L-methionine + thioredoxin disulfide; can act on free oxidized methionine with specificity for the R enantiomer; does not act on oxidized methionine in peptide linkage. Thioredoxin disulfide is the oxidized form of thioredoxin. (Gene Ontology, GO_0070191)