He accused the EU of wanting a solution that was "theoretically watertight" and unlikely to therefore ever be sustainable. Since the end of the EU transition period at the end of 2020, the EU has imposed strict customs checks on goods crossing the Irish Sea from Britain to Northern Ireland.
The bureaucratic checks were implemented as part of the 2019 Brexit withdrawal agreement but Unionists believe the heavy-handed nature of Brussels' approach is undermining the integrity of the UK.
Ministers have been in negotiations with the EU for months on the impact of the Protocol, but the bloc has so far refused to compromise.
Speaking to a parliamentary committee this morning, Mr Cleverly said ministers had solutions that should satisfy both sides in the talks.
He said: "There are a whole load of practical fixes which address the real world concerns of the EU in terms of integrity of the single market without introducing onerous or excessive checks or bureaucracy or delays in time with East-West trade across the Irish Sea between different parts of the United Kingdom."
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The minister said the proposal "also protects the North-South trade between the UK and Ireland".
He added: "We think we're there. We think we've got proposals that do it in the real world."
Both the UK and EU agreed in Brexit withdrawal talks not to impose customs checks on goods travelling between Northern Ireland and the Republic of Ireland in the south.
There were fears checks on the land border would undermine the Northern Ireland peace process and risk a return to the dark days of the troubles.
Brussels therefore demanded checks be carried out on goods crossing the Irish Sea in order to avoid products which do not comply with EU rules entering the bloc's single market via Northern Ireland.
The UK's plans would see customs checks eradicated on goods that were likely to remain in Northern Ireland and the introduction of a "trusted trader scheme" for big firms who pledge not to exploit the relaxing of the red tape.
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Any firm which abused the system would be hit with heavy fines.
Brussels has rejected the plans because it means a risk remains of a small number of goods could end up in the EU.
"Our position is we genuinely think that with a big dose of goodwill and trust and with a big dose of technology we can actually get through and address both our sets of concerns," Mr Cleverly added while giving evidence to the House of Lords Northern Ireland committee.
"It strikes me for the level of reassurance the EU seeks on the integrity of the single market, they are looking to put in processes which are not just practically but also theoretically water tight."
Britain has vowed to introduce legislation to override the Protocol unilaterally if the EU continues to refuse to negotiate.
Foreign Secretary Liz Truss unveiled the plans for the new law earlier this month and has pledged to start voting it through Parliament in a matter of weeks.
Ministers say they will continue to hold talks with the EU in parallel with the legislation.
They will abandon the new Bill if a compromise can be found with Brussels.
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'We have real world solutions!' EU blasted as Brussels makes unrealistic Brexit demands - Express
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