"Large parts of the [ministry of defense] and defense attaches around the world have been looking [for Russian and Soviet arms]," Wallace was quoted as saying by The Wall Street Journal.
The secretary noted that some 23 countries have Russian and Soviet arms in stock.
"If you really want to get it to the Ukrainians tomorrow morning so they can continue to fight... find what they're trained on, find what they need," Wallace said.
Wallace also mentioned that the British military repeatedly "bumped into the Russians looking, by the way, in some countries, looking also for some of their resupplies."
In an interview with the Washington Post, Wallace said it is unlikely that weapons delivered by the United Kingdom to Ukraine could be used to hit the Russian territory.
"First of all, we have not really given them weapons that probably could allow to do that. We have not given them helicopters or aircraft, or very long-range equipment, so it is unlikely British weapons could be used across the border," Wallace said, answering the question of whether the UK is fine with Ukraine using delivered arms to hit targets inside Russia.
The secretary noted that London supplies weapons to Ukraine in accordance with Article 51 of the UN Charter allowing countries to defend themselves.
"If they used British weapons, French weapons, German, anybody's weapons to achieve that effect, as long it is in accordance with international law, or humanitarian law, or Geneva Conventions, then, of course, that is something that we recognise as a low possibility because of type of weapons we provided, but nevertheless, it is a possibility because that was, the condition we gave it to the Ukrainians was to defend themselves," Wallace said.
Russia launched its special military operation in Ukraine on 24 February after the breakaway Donetsk and Lugansk people's republics appealed for help in fending off Kiev's provocations. The US and other Western countries have since been supplying Ukraine with lethal arms and pressuring Russia with sanctions.