The US would happily increase cooperation with Russia if only Russia would stop sabre rattling.
Actually, the USA has over 1,000 military bases abroad - many of them in Asia, as close as possible to the frontiers of Russia, China, and Iran. Not to mention its 11 (give or take) massive carrier battle groups which prowl the oceans to intimidate other nations.
Russia has, from memory, three or four small military bases outside its own borders. Apart from Syria, where it is fighting terrorism (because someone has to, and the USA isn't) they are all in friendly nations next door to Russia itself - Belarus, Armenia, etc.
When the USA sails its carrier battle groups or other naval units a few miles from Russia and China, in waters many thousands of miles away from the USA, that isn't sabre-rattling.
When the USA organizes "NATO" military exercises that involve large numbers of soldiers and weapons parading around a few miles from Russia, that isn't sabre-rattling.
When the USA places German tanks as close to St Petersburg as they were in autumn 1941, that isn't sabre-rattling.
When the USA places missile stations that could, without anyone knowing, be equipped to fire offensive nuclear missiles, within a few miles of Russia that isn't sabre-rattling. (Although it is exactly equivalent to the Soviet actions to prevent which President Kennedy precipitated the Cuban Missile Crisis).
But when the Russians or Chinese calmly announce the measures they have taken to defend themselves, or when they conduct military exercises entirely within their own borders - THAT is apparently sabre-rattling.
See http://www.globalfirepower.com.... Clue: Russia is the 5th-ranked in military spending, slightly below the UK. Its spending is less than one twelfth of the USA's.