Watson blew three match points as she lost 2-6 7-5 6-4 to qualifier Aleksandra Wozniak while fellow Brit Naomi Broady also bowed out in the opening round.
But Konta ensured there was home interest in the draw with a comfortable 6-3 6-1 win over Japan’s Kurumi Nara.
Elsewhere, top seed Ana Ivanovic and seventh seed Daniela Hantuchova both moved into the third round with straight set victories.
ATP Monte Carlo
Evans ends four-year wait to win on clay at Monte Carlo
Birmingham Classic results
Round Two
Lauren Davis (US) beat Victoria Duval (US) 6-2 2-6 6-2
7-Daniela Hantuchova (Slovakia) beat Belinda Bencic (Switzerland) 6-1 6-4
Coco Vandeweghe (US) beat Yanina Wickmayer (Belgium) 5-7 7-5 6-1
1-Ana Ivanovic (Serbia) beat Mona Barthel (Germany) 6-4 6-1
Round One
12-Monica Puig (Puerto Rico) beat Stefanie Voegele (Switzerland) 6-7(3) 6-2 6-2
Barbora Zahlavova Strycova (Czech Republic) beat Naomi Broady (Britain) 5-7 6-4 6-3
Aleksandra Wozniak (Canada) beat Heather Watson (Britain) 2-6 7-5 6-4
Timea Babos (Hungary) beat Marina Erakovic (New Zealand) 6-3 6-4
Christina Mchale (U.S.) beat Anna Karolina Schmiedlova (Slovakia) 7-6(5) 6-4
Johanna Konta (Britain) beat 14-Kurumi Nara (Japan) 6-3 6-1
Petra Cetkovska (Czech Republic) beat Alexandra Cadantu (Romania) 6-3 6-4
16-Casey Dellacqua (Australia) beat Urszula Radwanska (Poland) 6-1 6-3
Tokyo 2020
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ATP Monte Carlo
Medvedev withdraws from Monte Carlo after testing positive for Covid-19