Connors was much more successful against Borg in invitational events and exhibitions.
Invitational:
1976 Caracas Carpet RR Connors 6–4, 5–7, 6–3
1978 Suntory Cup, Tokyo Carpet F Borg 6–1, 6–2
1978 Buenos Aires Carpet F Connors 5–7, 6–3, 6–3
1979 Frankfurt Carpet F Borg 6–3, 4–6, 6–3, 6–4
1980 Europe vs. USA, Copenhagen Carpet RR Connors 6–4, 6–2
1980 Molson Light Challenge, Toronto Carpet RR Connors 7–5, 6–3
1982 Michelob Light Cup, Industry Hills Hard F Connors 5–7, 6–2, 6–2, 6–7(5–7), 6–2
1982 Molson Light Challenge, Montreal Carpet F Connors 6–4, 6–3
1982 Challenge of Champions, Chicago Carpet RR Borg 6–4, 1–6, 6–2
1983 Suntory Cup, Tokyo Carpet F Connors 6–3, 6–4
Bear in mind, there was a huge money at some of these events. I mentioned in another thread that the prize pot for the Suntory Cup in Tokyo was $250,000 for a 4 man invitational. The Wimbledon winner got $66,000 in the same year. They weren't knock about hits - serious money, serious intentions.