Fognini, who beat the Czech Republic's Jiri Vesely 7-6, 6-4, 6-2 to reach the third round, won his most recent meeting with Murray on his favourite surface of clay in Rome two months ago, but on grass the Scot will be the clear favourite.
But last year's quarter-finalist Pouille is out, the Frenchman beaten in four sets by Jerzy Janowicz on Court Three.
Germany's Brown, ranked No.97 in the world, came to the court with a reputation as a potential spoiler after knocking Nadal out in a second-round shock in 2015.
"It was a good match from my end", Murray said afterwards.
"I believe that I can have a good run here".
Murray added: "I've played against two guys with tough game styles but I did very well, I concentrated very well and put a lot of returns back in play".
"I really just tried to see the constructive and the positive things that I was doing and also accept the good things that she [Vekic] was doing, because she was doing very, very many good things". Murray has dropped just 14 games in his first two matches and is fresh off a victory that took a little over an hour and a half. I wouldn't say the target is to get five into the third round (Kyle Edmund, who faces Gael Monfils, could bring it up to that figure). But it was definitely a good one.
Jo-Wilfried Tsonga of France, the tournament number 12 seed, will battle Simone Bolelli of Italy for a spot in the third round, also at Court 2.
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The deal is thought to be separate to the one being negotiated for Wayne Rooney - who is heading in the opposite direction. Although Mourinho sold the Belgian while at Chelsea , he has been massively impressed by his development at Everton .
The 24th-seeded Querrey, who upset Novak Djokovic in last year's third round, defeated Basilashvili, from the country of Georgia, 6-4, 4-6, 6-3, 6-3 at the All England Club in London in one hour and 51 minutes. The other top 10-seeded man, Kei Nishikori, made it through Sergiu Stakhovsky, but needed four sets to do it, 6-4, 6-7 (7), 6-1, 7-6 (6).
"When the match (got) longer and longer, I felt a little bit sick and exhausted".
"I think the deeper it goes, the more hard it's going to be to beat him". "Both deserved to win".
"But I feel very fortunate to have had the encounters that I've had in this event over the years".
Azarenka recently returned to action after giving birth in December.
'I think I moved pretty well too.
Feliciano Lopez - one of the players singled out by Brown - was another to agree that the tactic had all but disappeared, although stopped short of blaming it all on the slower grass courts.
In the first round of The Championships, the world No.1 came up against youngster Alexander Bublik and came away with a straight sets, 6-1, 6-4, 6-2 victory.
"I look at it as continuous building and continuous work in every single match and practice that I have". The 4th-seeded Elina Svitolina of Ukraine tests Francesca Schiavone of Italy on Court 12 while the 18th-seed Dominika Cibulkova of Slovakia is set to meet Jennifer Brady of the United States of America on Court 18.