John Wombell, 58, from West Drayton, London, made the suggestion when he called Conservative MP Anna Soubry's constituency office on 30 November.
In June last year Labour MP Jo Cox was shot - a week before the EU referendum.
On Monday Wombell appeared at Central London Magistrates' Court and pleaded guilty to an offence under the Communications Act 2003.
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He pleaded guilty to one charge of sending a message that was grossly offensive or of an indecent, obscene or menacing nature.
He was also issued with a restraining order.
Following the hearing, Ms Soubry tweeted her thanks to Nottinghamshire and Met police forces for their investigation.
In the same week  as Wombell's threat, the Broxtowe MP, who has been vocal about being a Remain supporter, also received a misspelt tweet asking someone to "Jo Cox" her.
A police spokeswoman said a 25-year-old man arrested over the tweet to Mrs Soubry on 2 December had been released on bail.
The MP has made no further comments on the matter.
A man who called an MP's office and said she should be "Jo Cox'd" has been jailed for eight weeks.