How can a president make a agreement with the leaders of foreign country without involving Congress?

Answer 1

The president can confer with foreign officials and make agreements, sealed with a hand-shake. However, they are not binding unless they only involve something that is under control of the President. Foreign governments know how the US Constitution works and they realize that treaties have to be approved by the US Senate. They also keep track of political movements in the US and know what the Senate is likely to approve and what will face opposition.

There is also the possibility of secret deals but presidents who make them are treading dangerous ground if the secrets come out.