The human–animal relationship is an important welfare parameter in animal welfare assessment in cows, and the avoidance distance of cows to a stranger at the feed bunk is measured to assess this relationship. The assessment of the human–animal relationship in cow shelters in India, where old, unproductive, and abandoned cows are sheltered, is important to explore the welfare of cows in these shelters. The cows observed were of indigenous Indian breeds and breeds which were crosses between indigenous breeds and pure bred exotic cows. The human–animal contact in this context is of particular interest for welfare assessment as traditional Indian farming and sheltering systems involves regular close human–animal contact. In a cross-sectional study across 6 states, 54 cow shelters were visited and 30 cows in each shelter were randomly selected (1620 in total) for the assessment of avoidance distance and other cow-based (27 parameters) and resource-based (15 parameters) welfare parameters. Avoidance distance was assessed 1 h after morning feeding. Cows standing at the feeding manger were approached from the front at a rate of one step/s, starting 2 m away from the manger. The distance between the assessor’s hand and the cow’s head was estimated at the moment the cow moved away and turned its head, using a four-point scale (0, touched; 1, 0–50 cm; 2, 51–100 cm; and 3, >100 cm). The majority, 52%, of the cows allowed touch by the assessor and another 32% allowed approach within 50 cm, demonstrating tolerance, or even solicitation of close human–animal relationships by the cows. Avoidance distance increased with the proportion of cows with dirty hind limbs, tarsal joint swellings, and hair loss, and the extent of rumen fill. There was also evidence of reduced avoidance distances in cows with high levels of body condition score (BCS), dirty flanks, tarsal joint ulceration, carpal joint injuries, diarrhoea, hampered respiration, lesions on the body due to traumatic injuries, and body coat condition, probably as a result of moving difficulties. The avoidance distance was thus related to the health and welfare of the cows, providing a vital insight into the factors affecting human–animal contact in the shelters.