Simple Summary Consumers are increasingly aware of the nutritional quality of lamb products, especially in developed countries. Healthier lipid profiles might increase lamb meat consumption by concerned consumers. Pasture diets provide a viable option to enrich milk and meat products with fatty acids beneficial for human health. However, in Mediterranean areas, pasture is not available throughout the year, which means that weaned lambs are fed on concentrates. This investigation aimed to implement feeding strategies in suckling lamb to enhance healthier fatty acids in milk of dams and consequently in lamb’s meat by applying extruded linseed in a total mixed ration or using pasture. The proposed feeding plans were suitable to increase the n-3 fatty acids (FA) profile in milk and thus the lamb’s meat sourced from fresh pasture and linseed-enriched diets. Indoor rearing could include feeding lambs with linseed to help maintain a high level of beneficial fatty acids in lamb meat better than an un-supplemented diet or when pasture is not available. Abstract Consumers increasingly pay more attention to the lipid profile of meat products and consume less meat to reduce cholesterol and heart disease. In Italy, sheep producers are increasingly feeding sheep fresh forage. We investigated whether the supplementation of dam diet with extruded linseed would be an alternative strategy to pasture for improving the intramuscular and subcutaneous FA compositions of their suckling lambs. The ewe diets were enriched with either extruded linseed (L), un-supplemented farm diet (F), or pasture (P). Milk saturated fatty acids (SFA) decreased in P and L compared with F, while the opposite pattern was observed for polyunsaturated FA (PUFA) and conjugated linoleic acids after seven days. The FA composition of lamb meat was similar to that of their dam’s milk, showing higher PUFA in P and L compared to F, while SFA was higher in F. Regarding the lamb meat obtained from barn-held ewes, L had lower n-6/n-3 content compared to F, while an intermediate content was found in P. These results indicate a better n-3 FA profile in milk and lamb’s meat from pasture and linseed-enriched diets. No changes in lamb performance were observed.